In this paper, we bring together design theory and projective geometry over a ring. Both disciplines are well established, but the results on the interaction between them are rare and are scattered over the literature. 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.

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Divisible designs, Laguerre geometry, and beyond

Jordan homomorphisms connect directly with the structure of certain nonassociative rings including the classical Jordan algebras. The concept of a Jordan homomorphism generalizes in a natural way the ones of homomorphism and antihomomorphism of rings: in fact, by a well-known theorem of HUA'S [7], Jordan homomorphisms of (non-necessarily commutative) fields, and more generally Jordan homomorphisms onto prime rings [5], have to be homomorphisms or anti-homomorphisms of rings. Jordan homomorphisms also have an interest in Projective Geometry, where they can describe homomorphisms of geometric structures: to be exact, any Jordan homomorphism of an arbitrary field F, hence any automorphism or anti-automorphism of F, extends to a permutation of points of the underlying projective line by a direct action on the coordinates of any point. It is well-known that the induced permutation preserves the intrinsic geometric structure of the line. This structure is either the linear one inherited from an embedding into a projective space of higher dimension, in characteristic =~2 we then have harmonic quadruples (see [1], pp. 71--93), or is the non-linear one determined by a suitable subfield K, and we have a geometry of chains, a M6bius plane if [F : K] = 2 and F is commutative (see [4]). In recent years many authors, following a suggestion of R. BAER, have tried to verify whether this situation also occurs or not when passing to the more general case of an arbitrary (associative) ring, or of an arbitrary K-algebra for chain geometries. To verify that a Jordan homomorphism induces in the underlying projective line a well-defined mapping is then very difficult. Nevertheless for the case of a local ring, where the points have a very simple representation, the result has been achieved in [8], for the linear case, and in [6] for the non-linear case (here only for finite dimensional local K-algebras, K in the centre of the algebras). However, it is possible that, in the general case, there exist examples of Jordan homomorphisms which do not preserve the linear structure of the projective line: in [2], p. 367, we can infer examples which do not preserve a structure stronger than the linear one, defined by generalized harmonic quadruples. On the other hand, a Jordan homomorphism of finite dimensional K-algebras, containing K in their centre, has to be K-semilinear if it induces an homomorphism of the underlying chain geometries [6]. This is not true, of course, if K is not central,

Jordan homomorphisms, chain geometries and the fundamental theorem

B . A rtm a n n , G . D o rn , D . D rak e, G . T S rn er: H jelm slev 'sch e In zid en zg eo m etrie u n d v er- w an d te G e b ie te - L iteratu rv erzeich n is. J. G eo m . 7 (1 9 7 6 ), 1 7 5 -1 9 1 . D . Ju n g n ick el: H jelm slev 'sch e In zid en zg eo m etrie u n d v erw an d te G eb iete - L iteratu rv er- zeich n is II. J. G eo m . 1 6 (1 9 8 1 ), 1 3 8 -1 4 7 . A rticles o n reflectio n g eo m etry in th e sen se o f F . B a c h m a n n 's b o o k "A u fb a u d er G eo m etrie au s d em S p ieg elu n g sb eg riff", an d o n ch ain g eo m etries in th e sen se o f W . B en z' "G eo m etrie d er A lg eb ren " are n o t in clu d ed . P ap ers o n g eo m etric alg eb ra o v er rin g s (stru ctu re an d au to m o rp h ism s o f classical g ro u p s o v er rin g s) h av e also b een o m itted w ith so m e ex cep tio n s; fo r sy stem atic listin g s o f su ch p ap ers w e refer to M cD o n ald 76 an d H ah n ; O 'M e a ra 89. T h e au th o rs w ill b e g ratefu l fo r ad d itio n s, co m p letio n s an d co rrectio n s to th is list. A N D R I~ , J. : 69 l)b er H o m o m o rp h ism en p ro jek tiv er E b en en . A bh. M a th . Sere. Univ. H a m b u rg 3 4 (1969), 98-114. 72 U b er v erM lg em ein erte T ran slatio n sstru k tu ren u n d H o m o m o rp h ism en . A rch . M a th . (B a sel) 2 3 (1972), 198-205. AP~CHBOLD, J.W . : 55 P ro jectiv e g eo m etry o v er an alg eb ra. M a th em a tika 2 (1955), 105-115.

Literature on geometry over rings

This paper is the first part of a systematic survey on the structure of classical groups over general rings. We intend to cover various proofs of the main structure theorems, commutator formulas, finiteness and stability conditions, stability and prestability theorems, the nilpotency of K
 1, the centrality of K
 2, automorphisms and homomorphisms, etc. This first part covers background material such as one-sided inverse, elementary transformations, definitions of obvious subgroups, Bruhat and Gaus decompositions, relative subgroups, finitary phenomena, and transvections. Bibliography: 674 titles.

Linear groups over general rings. I. Generalities

We determine all distant-isomorphisms between projective lines over semilocal rings. In particular, for those semisimple rings that do not have a simple component which is isomorphic to a field, every distant isomorphism arises from a Jordan isomorphism of rings and a projectivity. We show this by virtue of a one-one correspondence linking the projective line over a semisimple ring with a Segre product of Grassmann spaces.

On distant-isomorphisms of projective lines

E. ARTIN (1957), R. BAER (1952) and J.DIEUDONNE (1951) emphasized many times the structural identity between classical projective geometry and linear algebra over a division ring. Then Baer pointed out a possible extension of this structural identity to the case of a ring, generating intense research activity in the area of geometric algebra over rings. In this direction the most significant results appear: Ojanguren and Sridharan’s article on the fundamental theorem of projective geometry over a commutative ring; some theorems due to Klingenberg, Bass and Suslin on the structure of the general linear group over appropriate rings and some valuable notes of O’Meara’s on the automorphisms of linear groups over an integral domain.

Topics in Geometric Algebra over Rings

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Topics in Geometric Algebra over Rings