Generatrices and relations of a generalized orthogonal group over commutative semilocal rings without identity

Abstract

The description of orthogonal groups (and close to them ones) in terms of generatrices and relations represents one of the main problems of the combinatorial theory of linear groups. This paper is dedicated to the mentioned question; namely, here we find generating elements and defining relations of a generalized orthogonal group O◦(n,R), n ≥ 2, over a commutative semilocal ring R (in general, without identity) under certain natural constraints. In order to formulate the problem exactly, let us define several necessary notions. LetΛ be an arbitrary associative ring and let ◦ be its quasi-multiplication (i. e., x ◦ y = x+ xy + y). An element α from Λ is said to be quasi-invertible, if with certain α′ ∈ Λ, α ◦ α′ = 0 = α′ ◦ α. If α is quasi-invertible, then one can uniquely determine its quasi-inverse α′. The set of all quasi-invertible elements Λ◦ from Λ forms a group with respect to the composition ◦ (where the identity is zero). We consider the case when Λ = M(n,R) is a complete matrix ring over a ring T which is (associative-)commutative and not necessarily with identity. Let mean the transposition in Λ. The set of quasi-invertible matrices from Λ such that a′ = a forms a group with respect to the matrix quasi-multiplication. We denote it by O◦(n,R) and call it a generalized orthogonal group of the degree n over the ring R. Note that if R contains the identity, then the mapping O(n,R) → O◦(n,R), E + a → a, where E is the unit matrix of the order n, defines an isomorphism. So the introduced groupO◦(n,R) generalizes the notion of a usual orthogonal group to the most general cases of associative-commutative rings R. Let now R be a commutative semilocal ring (not necessarily with identity) and let J = J(R) be its Jacobson radical. By definition it means that R/J ∼= k1 ⊕ · · · ⊕ km, where ki are certain fields (i = 1, . . . ,m). Let Ri stand for the complete preimage of the addend ki under the natural epimorphism R → R = R/J, x → x = x+ J. (1)