Divisible Designs, Laguerre Geometry, and Beyond
Contents
Chapter 1 Introduction

This is a revised and updated version of our lectures notes [60] from the Summer School on Combinatorial Geometry and Optimisation 2004 “Giuseppe Tallini” which took place at the Catholic University of Brescia, Italy.
In these notes we aim at bringing together design theory and projective geometry over a ring. Both disciplines are well established, but the results on the interaction between them seem to be rare and scattered over the literature. Thus our main goal is to present the basics from either side, to develop, or at least sketch, the principal connections between them, and to make recommendations for further reading. There is no attempt to provide encyclopedic coverage with expansive notes and references.
In Chapter 2 we start from the scratch with divisible designs. Loosely speaking, a divisible design is a finite set of points which is endowed with an equivalence relation and a family of distinguished subsets, called blocks, such that no two distinct points of a block are equivalent. Furthermore, there have to be several constants, called the parameters of the divisible design, as they govern the basic combinatorial properties of such a structure. Our exposition includes a lot of simple examples. Also, we collect some facts about group actions. This leads us to a general construction principle for divisible designs, due to Spera. This will be our main tool in the subsequent chapters.
Next, in Chapter 3 we take a big step by looking at the classical Laguerre geometry over the reals. This part of the text is intended mainly as a motivation and an invitation for further reading. Then we introduce our essential geometric concept, the projective line over a ring. Although we shall be interested in finite rings only, we do not exclude the infinite case. In fact, a restriction to finite rings would hardly simplify our exposition. From a ring containing a field, as a subring, we obtain a chain geometry. Again, we take a very short look at some classical examples, like Möbius geometries. Up to this point the connections with divisible designs may seem vague. However, if we restrict ourselves to finite local rings then all the prerequisites needed for constructing a divisible design are suddenly available, due to the presence of a unique maximal ideal in a local ring.
Chapter 4 is entirely devoted to the construction of a divisible design from the projective line over a finite local ring. The particular case of a local algebra is discussed in detail, but little seems to be known about the case of an arbitrary finite local ring, even though such rings are ubiquitous. It is worth noting that the isomorphisms between certain divisible design can be described in terms of Jordan isomorphisms of rings and projectivities; strictly speaking this applies to divisible designs which stem from chain geometries over local algebras with sufficiently large ground fields. Geometric mappings arising from Jordan homomorphisms are rather involved, and the related proofs have the tendency to be very technical; we therefore present this material without giving a proof.
Chapter 5 can be considered as an outlook combined with an invitation for further research. We sketch how one can obtain an equivalence relation on the projective line over any ring via the Jacobson radical of the ring. Recall that such an equivalence relation is one of the ingredients for a divisible design. The maximal ideal of a local ring is its Jacobson radical, so that we can generalise some of our results from a local to an arbitrary ring. It remains open, however, if this equivalence relation could be used to construct successfully a divisible design even when the ring is not local. Finally, we collect some facts about finite chain geometries. Their combinatorial properties are—in a certain sense—almost those of divisible designs, but no systematic treatment seems to be known.
Chapter 2 Divisible Designs
2.1 Basic concepts and first examples
2.1.1.
Suppose that a tournament is to take place with participants coming from various teams, each team having the same number of members, say . In order to avoid trivialities, we assume and . So there are teams. The tournament consists of a number of games. In any game participants from different teams play against each other. Of course, there should be at least two teams, i. e., .
The problem is to organdie this tournament in such a way that all participants are “treated equally”. Strictly speaking, the objective is as follows:
The number of games in which any two members from different teams play against each other has to be a constant value, say .
In this way it is impossible that one participant would have the advantage of playing over and over again against a small number of members from other teams, whereas others would face many different counterparts during the games.
In the terminology to be introduced below, this problem amounts to constructing a divisible design with elements. The points of the divisible design are the participants, the point classes are the teams, and the blocks correspond to the games. Many of our examples will give solutions to this problem for certain values of , , , and .
2.1.2.
Throughout this chapter we adopt the following assumptions: is a finite set with an equivalence relation . We denote by the equivalence class of and define
(2.1) 
A subset of is called transversal if for all . Observe that here the word “transversal” appears in a rather unusual context, since it is not demanded that meets all equivalence classes in precisely one element. Cf., however, the definition of a transversal divisible design in 2.1.5.
Definition 2.1.3.
A triple is called a divisible design if there exist positive integers such that the following axioms hold:

is a set of transversal subsets of with for all .

for all .

For each transversal subset there exist exactly elements of containing .

, where .
The elements of are called points, those of blocks, and the elements of point classes.

We shall frequently use the shorthand ‘‘DD’’ for “divisible design”. Sometimes we shall speak of a DD without explicitly mentioning the remaining parameters , , and . According to our definition, a block is merely a subset of . Hence the DDs which we are going to discuss are simple, i. e., we do not take into account the possibility of “repeated blocks”. Cf. [14, p. 2] for that concept.
Since is determined by and vice versa, we shall sometimes also write a divisible design in the form rather than .
2.1.4.
Let us write down some basic properties of a DD. Since , axiom (D) implies that
(2.2) 
or, said differently, that . From this and (B) we infer that
(2.3) 
Hence, by (D) and (B), there exists at least one transversal subset of , say . By virtue of (C), this is contained in blocks so that
(2.4) 
So, since , we can derive from axiom (A) and (2.3) the inequality
(2.5) 
2.1.5.
A divisible design is called transversal if each block meets all point classes, otherwise it is called regular. Hence a DD is transversal if, and only if equality holds in (2.5).
During the last decades there has been a change of terminology. Originally, the point classes of a DD were called point groups and DDs carried the name groupdivisible designs. In order to avoid confusion with the algebraic term “group”, in [13] this name was changed to read groopdivisible designs. We shall not use any of these phrases.
2.1.6.
Let us add in passing that some authors use slightly different axioms for a DD in order to exclude certain cases that do not deserve interest. For example, according to our definition is allowed, but this forces .
On the other hand, our axiom (D) is essential in order to rule out trivial cases which would cause a lot of trouble. If we would allow then there would not be any transversal subset of , and (C) would hold in a trivial manner. Such a value for would therefore have no meaning at all for a structure .
Examples 2.1.7.
We present some examples of DDs.

We consider the Pappos configuration in the real projective plane which is formed by points and lines according to Figure 2.1.
We obtain a DD, say , as follows: Let
i. e., . The blocks are, by definition, the subsets of collinear points in , so that . We define three point classes, namely , , and , each with elements. Then for any two points from distinct point classes are is a unique block containing them. So and . This DD is transversal.

Let us take a regular octahedron in the Euclidean space (Figure 2.2), and let us turn it into a DD as follows:
Denote by the set of all vertices of the octahedron. For all we put if, and only if, and are opposite vertices. Hence . The blocks are defined as the triangular faces, whence . So we get a transversal divisible design.

Our next example is the projective plane of order three which is depicted on the left hand side of Figure 2.3. It is a DD with points. There are blocks; they are given by those subsets of the point set which consist of points on a common curve. (Some of these curves are segments, others are not.) There are point classes, because means that all point classes are singletons.
We shall not need the definition of a finite projective plane and refer to [14, p. 6]. Let us add, however, that in the theory of projective planes one speaks of lines rather than blocks. The order of a projective plane is defined to be if there are points on one (or, equivalently, on every) line.
Let us remove one point from the point set of this projective plane. Also, let us redefine the point classes as the four truncated lines (illustrated by thick segments and a thick circular arc), the other nine lines remain as blocks. This yields a DD.
If we delete one line and all its points from the projective plane of order three then we obtain the affine plane of order three. Each of the twelve remaining lines gives rise to a block with three points, the point classes are defined as singletons. As before, one speaks of (affine) lines rather than blocks in the context of affine planes. Observe that the order of an affine plane is just the number of points on one (or, equivalently, on every) line. See [14, p. 8] for further details.
This affine plane is a DD with points and, as before, all point classes are singletons. See the third picture in Figure 2.3. Two lines of an affine plane are called parallel if they are identical or if they have no point in common.
Finally, we change the set of lines and the set of point classes of this affine plane as follows: We exclude three mutually parallel lines from the line set, turn them into point classes, and disregard the oneelement point classes of the underlying affine plane. The remaining nine lines are considered as blocks. In this way a DD with points is obtained. On the right hand side of Figure 2.3 the bold vertical segments represent the point classes.

We proceed as in the previous example, but starting with the projective plane of order two which is a DD with points. In this way we obtain a DD with points, a DD with points (the affine plane of order ), and a DD with points. See Figure 2.4.

It is easy to check that the DD from Example (b) is also a DD; likewise all our DDs are at the same time DDs. Thus the previous examples illustrate the following result:
Theorem 2.1.8.
Let be a DD with and let be an integer such that . Then is also an DD with
(2.6) 
Proof.
We fix one transversal subset . The proof will be accomplished by counting in two ways the number of pairs , where is a subset of such that is a transversal subset, and where is a block containing .
On the one hand, let us single out one of the blocks containing . Then there are
possibilities to choose a within that particular block.
On the other hand, to select an arbitrary amounts to the following: First choose point classes out of the point classes that are disjoint from (cf. (2.3)), and then choose in each of these point classes a single point (out of ). Hence there are precisely
ways to find such a . For every there are pairs with the required property.
Altogether we obtain
(2.7) 
which completes the proof. ∎
2.1.9.
Theorem 2.1.8 enables us to calculate several other parameters of a DD. Letting in formula (2.6) provides the number of blocks, i. e.
(2.8) 
Likewise, for we obtain the number
(2.9) 
of blocks through a point which is therefore a constant. Provided that formula (2.6) reads
(2.10) 
By Theorem 2.1.8, formula (2.10) remains valid if is replaced with an integer , subject to the condition . Hence we infer the equation
(2.11) 
by letting . For we may let which gives
(2.12) 
The last two equations are just particular cases of formula (2.7).
2.1.10.
A divisible design with is called a design; we refer to [48], [80], [89], or the two volumes [14] and [15]. In design theory the parameter is not taken into account, and a DD with points is often called a design. Of course, this is a different notation and we urge the reader not to draw the erroneous conclusion “” when comparing these lecture notes with a book on design theory.
2.1.11.
If is a DD and is a DD then an isomorphism is a bijection
such that
(2.13)  
(2.14) 
Clearly, the inverse mapping of an isomorphism is again an isomorphism. If the product of two isomorphisms is defined (as a mapping) then it is an isomorphism. The set of all isomorphisms of a DD onto itself, i. e. the set of all automorphisms, is a group under composition of mappings.
2.1.12.
Suppose that there exists an isomorphism of a DD onto a DD . Such DDs are said to be isomorphic. Then
However, in view of Theorem 2.1.8 we may have . Thus we impose the extra condition that the parameters and are maximal, i. e., is a DD but not a DD, and likewise for . Then, clearly,
2.1.13.
Condition (2.14) in the definition of an isomorphism can be replaced with the seemingly weaker but nevertheless equivalent condition
(2.15) 
Suppose that we are given a bijection satisfying (2.15). If for some subset of then there is an . Hence with by (2.15). Since two equivalence classes with a common element are identical, we get and, finally, . In sharp contrast to this result, the equivalence sign in (2.13) is essential. Cf. Example 2.1.14 below.
Example 2.1.14.
Let us consider once more a regular octahedron in the Euclidean space. We turn the set of its vertices into a DD with points in two different ways (Figure 2.5): For both DDs the point classes are the sets of opposite vertices. However, the blocks are different. Firstly, we take all triangular faces as blocks (left image). This gives a DD which is also a DD. Cf. Example 2.1.7 (b). Secondly, only triangular faces (given by the shaded triangles in the right image) are considered as blocks, so that a DD is obtained.
Observe that the identity mapping maps every block of the second design onto a block of the first design, but not vice versa. Hence a bijection between the point sets of DDs which preserves point classes in both directions and blocks in one direction only, need not be an isomorphism.
2.2 Group actions
2.2.1.
Let us recall that all bijections (or permutations) of a finite set^{1}^{1}1Most of the results from this section remain true for an infinite set . form the symmetric group . If is any group then a homomorphism
is called a permutation representation of . In this case the group is also said to operate or act on via . In fact, each yields the bijection
Whenever is clear from the context, then we shall write for the image of under the permutation . Thus, if the composition in is written multiplicatively, we obtain
Provided that is injective the representation is called faithful. So for a faithful representation we have as is kernel, and we can identify with its image . However, in most of our examples the representation will not be faithful, i. e., there will be distinct elements of which yield the same permutation on .
2.2.2.
For the remaining part of this section we suppose that acts on (via ).
For each we write for the orbit of under . The set of all such orbits is a partition of . If itself is an orbit then is said to operate transitively on . This means that for any two elements there is at least one with . If, moreover, this is always uniquely determined then the action of is called regular or sharply transitive. If operates regularly on then the representation is necessarily faithful, since every has the property for all , whence .
The given group acts also in a natural way on certain other sets which are associated with . E.g., for every nonnegative integer , the group acts on the fold product by
If this is a transitive action on the subset of tuples with distinct entries from then one says that acts transitively on .
Moreover, for , the group acts on the (non empty) set of all subsets of by
In case that this is a transitive action, the group is said to act homogeneously on .
Similarly, acts on the power set of .
Later, we shall be concerned with homogeneous and transitive group actions. Thus the following result, due to Donald Livingstone and Ascher Wagner [90], deserves our interest, even though we are not going to use it.
Theorem 2.2.3.
Suppose that the action of a group on a finite set is homogeneous, where . Then acts transitively on . If, moreover, then even acts transitively on .
2.2.4.
An equivalence relation on is called invariant if
(2.16) 
Then
(2.17) 
follows immediately, by applying (2.16) to and . The finest and the coarsest equivalence relation on , i. e. the diagonal and , obviously are invariant equivalence relations on .
Suppose now that acts transitively on . If and are the only invariant equivalence relations on then the action of is said to be primitive; otherwise the action of is called imprimitive.
Suppose that acts imprimitively on . A subset is called a block of imprimitivity if it is an equivalence class of a invariant equivalence relation, say , which is neither nor . Thus a block of imprimitivity is a subset of such that , , and for all we have either or .
2.2.5.
Given a subset the setwise stabiliser of in is the set , say, of all satisfying . This stabiliser is a subgroup of . The pointwise stabiliser of in is the set of all such that for all . This pointwise stabiliser is also a subgroup of and, clearly, it is a normal subgroup of the setwise stabiliser .
If then we simply write instead of . With this convention, the mapping is a bijection of the orbit onto the set of right cosets of in , whence we obtain the fundamental formula
(2.18) 
It links cardinality of the orbit with the index of the stabiliser in , i. e. the number of right (or left) cosets of in .
We refer to [82, pp. 71–79] for a more systematic account on group actions.
2.3 A theorem of Spera
2.3.1.
One possibility to construct divisible designs is given by the following Theorem which is due to Antonino Giorgio Spera [118, Proposition 3.2]. A similar construction for designs can be found in [14, Proposition 4.6].
The ingredients for this construction are a finite set with an equivalence relation on its elements, a finite group acting on , and a socalled base block (or starter block) , say. Its orbit under the action of will then be our set of blocks. More precisely, we can show the following:
Theorem 2.3.2.
Let be a finite set which is endowed with an equivalence relation ; the corresponding partition is denoted by . Suppose, moreover, that is a group acting on , and assume that the following properties hold:

The equivalence relation is invariant.

All equivalence classes of have the same cardinality, say .

The group acts transitively on the set of transversal subsets of for some positive integer .
Finally, let be an transversal subset of with . Then
is a divisible design, where
(2.19) 
and where denotes the setwise stabiliser of .
Proof.
Firstly, let . Since is transversal, we have so that axiom (D) in the definition of a DD is satisfied. Also, we obtain .
As is an transversal set, so is every element of by (2.17). This verifies axiom (A), whereas axiom (B) is trivially true due to assumption (b).
Next, to show axiom (C), we consider the base block and a subset which exists due to our assumption . Let be the number of blocks containing . Given an arbitrary transversal subset there is a with , since is transversal. This takes the distinct blocks through to distinct blocks through . Similarly, the action of shows that there cannot be more than blocks containing .

Note that in [118] our condition (b) is missing. On the other hand it is very easy to show that (b) cannot be dropped without effecting the assertion of the theorem:
Example 2.3.3.
Let , , and let be that subgroup of the symmetric group which is formed by the identity and the transposition that interchanges with . Then, apart from (b), all other assumptions of Theorem 2.3.2 are satisfied if we define and . However, no DD is obtained, since there are two blocks containing , but there exists only one block through the point .
2.3.4.
In the subsequent chapters we shall mainly apply a slightly modified version of Theorem 2.3.2 which is based on the following concept. A tuple is called transversal if its entries belong to distinct point classes.
Corollary 2.3.5.
Theorem 2.3.2 remains true, mutatis mutandis, if assumption (b) is dropped and assumption (c) is replaced with

The group acts transitively on the set of transversal tuples of for some positive integer .
Proof.
We observe that each transversal subset gives rise to mutually distinct transversal tuples with entries from . As , it is obvious from (c) that acts transitively on the set of transversal subsets of , i. e., condition (c) from Theorem 2.3.2 is satisfied.
In order to show that all equivalence classes of are of the same size, we prove that acts transitively on . Since assumption (a) remained unchanged, formula (2.17) can be shown as before. This implies that, for all and all , the image is an equivalence class; hence acts on . For this action to be transitive it suffices to establish that operates transitively on . So let and be arbitrary elements of . We infer from that there exist transversal tuples and . By (c), there is at least one which takes the first to the second tuple. Therefore . ∎
2.3.6.
2.3.7.
Clearly, Theorem 2.3.2 remains valid if we replace assumption (b) with the following:

acts transitively on .
Another possibility to alter the conditions in Theorem 2.3.2 is as follows [111, Remark 2.1]: Suppose that condition (b) is dropped and that (c) is replaced with

The group acts transitively on the set of transversal subsets of for some positive integer ..
In this case, let , where , be a system of representatives for the equivalence classes of such that . We claim that (c) implies
which in turn is equivalent to (b). By (c), we have so that
are transversal subsets of . By the action of on , the tuple
arises from
by rearranging its entries. Therefore we obtain , as required.
Finally, we may even just drop assumption (b) if the integer admits the application of Theorem 2.2.3 which in turn will ensure that acts transitively on .
2.4 Divisible designs and constant weight codes
2.4.1.
There is a close relationship between DDs and certain codes which will be sketched in this section.
First, we collect some basic notions from coding theory. See, among others, the book [75] for an introduction to this subject. Let us write^{2}^{2}2In Chapter 3 we shall use this symbol to denote the ring of integers modulo .
Also let be a positive integer. The Hamming distance of and is defined as the number of indices such that . It turns into a metric space. The Hamming weight of an element is its Hamming distance from or, said differently, the number of its nonzero entries. This terminology is in honour of Richard Wesley Hamming (1915–1998), whose fundamental paper on errordetecting and errorcorrecting codes appeared in 1950.
For our purposes it will be adequate to define an automorphism of as a product of any two mappings of the following form: First we apply a bijection
where each is a permutation of , and then a bijection
where is a permutation of . All such automorphisms form a group under composition of mappings. Every automorphism preserves the Hamming distance. The Hamming weight is preserved if, and only if, remains fixed.
An ary code of length is just a given subset . Its elements are called codewords. The set is called the underlying alphabet of the code . A code is called a constant weight code if all codewords have the same (constant) Hamming weight.
Let be codes. An isomorphism is an automorphism of taking to . An automorphism of a code is defined similarly.
2.4.2.
We now present the essential construction: Suppose that is a DD with point classes. Also let . We augment ideal points to , thus obtaining a set with
To each point class we add precisely one ideal point in such a way that distinct point classes are extended by distinct ideal points. Given a point class we write for the corresponding extended point class. Any block has points. We turn it into an extended block, say , by adding to the ideal points of those extended point classes which have empty intersection with . Hence meets every extended point class at precisely one point.
By the above, there exists a bijection
such that for each point class there is an index with
This means that under the set of ideal points goes over to . Furthermore, two points of are in the same extended point class if, and only if, the first entries of their images coincide.
We are now in a position to define the code of (with respect to ) as the subset of given by
According to our construction, all codewords have weight , whence is in fact a constant weight code.
In general, can be chosen in different ways. However, this will yield isomorphic codes. So the actual choice of turns out to be immaterial. In [113] the codes arising in this way are characterised. Also, it is shown that the entire construction can be reversed, i. e., one can go back from certain codes to divisible designs.
2.4.3.
A neat connection exists between the automorphism group of a DD and the automorphism group of its constant weight code. Up to the exceptional case when and , the two groups are isomorphic [113, Theorem 3.1]. Also, if the automorphism group of is “large” then its corresponding code is well understood. See [51], [109], and [113] for a detailed discussion.
2.5 Notes and further references
2.5.1.
There is a widespread literature on divisible designs, and some particular classes of DDs have been thoroughly investigated and characterised.
Among them are translation divisible designs, i. e. DDs with a group of automorphisms which acts sharply transitive on (see 2.2.2) such that the following holds: For all blocks and all there is either or . The name of these structures is due to the fact the same properties hold, mutatis mutandis, for the action of the group of translations on the set of points and lines of the Euclidean plane. We refer to [16], [74], [83], [103], [104], [105], [106], [107], [108], [115], [116], and the references given there.
Another construction of these more general DDs uses a Singer group with a relative difference set [84]. As a general theme, each of the preceding constructions is based upon a group which acts as a group of automorphisms of the DD.
2.5.2.
While Theorem 2.3.2 and Corollary 2.3.5 pave the way to constructing DDs, the actual choice of , , , and a base block is a subtler question. We collect here some results:
In [118] the following case is considered: is the projective line over a finite local algebra , and is the general linear group in two variables over . All this is part of our exposition in Chapters 3 and 4. In this way one obtains divisible designs.
A higherdimensional analogue, based upon the projective space over a finite local algebra can be found in [119]; here, in general, only DDs are obtained.
Another approach uses as the set the set of (affine) lines of a finite translation plane, is chosen to be the usual parallelism of lines, and is a group of affine collineations which acts transitively on the line at infinity and contains all translations. Apart from the finite Desarguesian planes this leads to Lüneburg planes and Suzuki groups; see [112] and [121]. A more general setting, where acts transitively on a subset of the line at infinity can be found in the papers [43], [46], and [111].
A class of DDs, where is an orthogonal group or a unitary group, is determined in [45]. It was pointed out in [55] that one particular case of this construction is—up to isomorphism—a Laguerre geometry (see 3.5.13) which, by a completely different approach, appears already in [118].
In [44] the group is chosen to be the classical group (the general linear group in variables over the field with elements) in order to obtain divisible designs.
Also, we refer to [120] for a discussion of transitive extensions of imprimitive groups. A generalisation of Spera’s construction was exhibited in [54] and [56]. It was put into a more general context in [38] as follows: Let a group acting on some set and a starter DD in be given. Then, under certain technical conditions, a new DD can be obtained via the action of on .
Chapter 3 Laguerre Geometry
3.1 Real Laguerre geometry
3.1.1.
The classical Laguerre geometry is the geometry of spears and cycles in the Euclidean plane. A spear is an oriented line and a cycle is either an oriented circle or a point (a “circle with radius zero”). There is a tangency relation between spears and cycles; see the first two images in Figure 3.1. Furthermore, there exists a parallelism (written as ) on the set of spears which is depicted in the third image. We shall not give formal definitions of these relations.
For our purposes it is more appropriate to identify a cycle with the set of all its tangent spears. Then it is intuitively obvious that any cycle contains precisely one spear from every parallel class, i. e., it is a “transversal set”. Also, given any three nonparallel spears there is a unique cycle containing them. All this reminds us of a divisible design, even though the set of spears is infinite.
3.1.2.
It was in the year of 1910 that Wilhelm Blaschke (1885–1962) showed that the set of spears is in oneone correspondence with the points of a circular cylinder of the Euclidean space [17], now called the Blaschke cylinder^{1}^{1}1From the point of view of projective geometry this is a quadratic cone without its vertex, whence it is also called the Blaschke cone.. Under this mapping the cycles correspond to the ellipses on the cylinder and two spears are parallel if, and only if, their images are on a common generator of the cylinder. Blaschke also showed that the real Laguerre geometry can be represented in terms of dual numbers , where , , and ; see Example 3.5.4 (b) for a concise definition.
3.1.3.
There is a wealth of literature on the classical Laguerre geometry. We refer to [5, Chapter 1,§ 2], [6, Chapter 4], [53, Chapter 15 A], [99], [129], [130], as well as the survey articles [67] and [102]. Note that in [129] the term inversive Galileian plane—named after Galileo Galilei (1564–1642)—is used instead.
3.2 The affine and the projective line over a ring

All our rings are associative with a unit element (usually denoted by ), which is inherited by subrings and acts unitally on modules. The trivial case is excluded.
3.2.1.
Let be a ring. Given an element there are various possibilities:
If there is an with then is called left invertible . Such an element is said to be a left inverse of . Right invertible elements and right inverses are defined analogously.
If has both a left inverse and a right inverse then
(3.1) 
In this case, the element is said to be invertible. Moreover, by the above, all left (right) inverses of are equal to () so that it is unambiguous to call the inverse of . The (multiplicative) group of invertible elements (units) of a ring will be denoted by . Clearly, is neither left nor right invertible.
If then is called a left zero divisor if there exists a nonzero element such that . Such an has no left inverse, since would imply . However, an element without a left inverse is in general not a left zero divisor. Right zero divisors are defined similarly.
Of course the distinction between “left” and “right” is superfluous if is a commutative ring.
3.2.2.
Suppose that we are given elements with . Hence for all . This implies that the right translation is surjective. Moreover, . Thus, whenever we are able to show that is injective we obtain , i. e., . This conclusion can be applied, for example, if is a finite ring or a subring of the endomorphism ring of a finitedimensional vector space.
Rings with the property that, for all , implies are called Dedekindfinite (see e.g. [86]). In fact, in most of our examples this condition will be satisfied. It carries the name of Richard Dedekind (1831–1916).
Exercise 3.2.3.
Show that the endomorphism ring of an infinite dimensional vector space is not Dedekindfinite.
3.2.4.
Let be a ring. Then it is fairly obvious how to define the affine line over . It is simply the set , but—as in real or complex analysis—we adopt a geometric point of view by using the term point for the elements of . We shall meet again this affine line as a subset of the projective line over . However, to define something like a “projective line” over a ring is a subtle task. As a matter of fact, various definitions have been used in the literature during the last decades. Some of those definitions are equivalent, some are equivalent only for certain classes of rings. A short survey on this topic is included in [88].
3.2.5.
Of course, a definition of a projective line over a ring has to include, as a particular case, the projective line over a field . Observe that we use the term “field” for what other authors call a skew field or a division ring. Thus multiplication in a field need not be commutative.
A particular case is well known from complex analysis: The complex projective line can be introduced as , where is an arbitrary new element. Intuitively, we think of as being , where is nonzero. For all , we have . Thus every fraction other than “” determines an element of . ,
It is immediate to carry this over to an arbitrary field . However, one has to be careful when using fractions in case that is noncommutative, since could mean or . We avoid ambiguity by representing the elements of the projective line over an arbitrary field via
(3.2) 
More formally, the projective line over appears as the set of onedimensional subspaces of the left vector space . Every nonzero vector in is a representative of a point. In terms of the projective line the zero vector has no meaning. Of course, we could also consider as a right vector space in order to describe this projective line. The choice of “left” or “right” is just a matter of taste.
3.2.6.
Now let us turn to an arbitrary ring . We consider a (unitary) left module over . A family of vectors in is called a basis provided that the mapping
(3.3) 
is a bijection. In this case is called free of rank . It is important to notice that this rank is in general not uniquely determined by . See, for example, [87, Example 1.4]
In order to define the projective line over a ring we start with a module over which is free of rank . By virtue of the bijection given in (3.3), we replace with . Of course, the left module is free of rank ; this is immediate by considering the standard basis of .
It is tempting to define the projective line over a ring just in same way as we did for a field in (3.2). However, this would not give “enough points”, since we would not get any “point” of the form , where has no left inverse. Nevertheless, would be a point, i. e., we would not have symmetry with respect to the order of coordinates. At the other extreme one could say, as in the case of a field, that every pair , should be a representative of some point. This point of view is adopted, for example, in [42, p. 1128], where a distinction between “points” and “free points” is made, and in [52]. Yet, also here a problem arises: By following this approach we would get, in general, “far too much points” for our purposes.
It turned out that a “good” definition of the projective line over a ring is as follows: A submodule is a point if is an element of a basis with two elements. As in the case of a vector space, the general linear group of invertible matrices with entries in acts regularly on the set of those ordered bases of which consist of two vectors. Therefore, starting at the canonical basis we are lead to the following strict definition:
Definition 3.2.7.
The projective line over is the orbit
of under the natural action of on the subsets of . Its elements are called points.
3.2.8.
Let us describe in different words: A pair is called admissible (over ) if there exist such that
(3.4) 
Thus our definition of the projective line relies on admissible pairs. However, there may also be nonadmissible pairs such that . Strictly speaking, this phenomenon occurs precisely when is not Dedekindfinite (see 3.2.2). We refer to [31], Propositions 2.1 and 2.2, for further details. We therefore adopt the following convention:
Points of are represented by admissible pairs only.
This brings us in a natural way to the next result:
Theorem 3.2.9.
Let and be admissible pairs. Then if, and only if, there exists an element with .
Proof.
Let . By our assumption, there is a matrix with first row . Thus
As is admissible, so is . Now implies for some , whence . Similarly, we obtain for some . This means that is a left inverse of . By the above, is admissible. Hence there exists an invertible matrix , say, with first row . Then
shows that , i. e. the northwest entry of , is a right inverse of . Therefore
as required.
Conversely, if is a unit with then , whence . ∎
3.2.10.
We note that, for all ,
(3.5) 
Hence the projective line over contains all points with . If are different then . Analogous results hold for for all . However, if then , i. e., this point is taken into account for a second time. This shows that we can restrict ourselves to points with , and it establishes the estimate
(3.6) 
We shall see below that for certain rings the projective line contains even more points. Cf. however Theorem 3.5.5 and Corollary 3.5.6.
Example 3.2.11.
Let be the (commutative) ring of integers modulo . We have , where ; the ideals of are , , , and . Cf. [82, 2.6] for further details.