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We prove a necessary condition and a sufficient condition for an n × n -matrix A with determinantal rank ρ (A) = t over an arbitrary commutative ring to be (von Neumann) strongly regular in terms of the trace of its t th compound matrix C t (A). In particular, a non-zero n × n -matrix A with ρ (A) = t over a local commutative ring R is strongly regular if and only if Tr (C t (A)) is a unit in R , and in this case we construct a strong inner inverse of A. We derive applications to products of local commutative rings and group algebras. Finally, we count strongly regular matrices over some finite rings of residue classes and group algebras. [ABSTRACT FROM AUTHOR]

We investigate the structure of possibly degenerate ε -hermitian forms over local rings. We prove classification theorems in the cases where the ring is complete and either the form is nondegenerate or the ring is a discrete valuation ring. In the latter case we describe a complete set of invariants for such forms based on a generalisation of the classical notion of the radical of the form. [ABSTRACT FROM AUTHOR]

Abstract: Let K s(R) be the generalized matrix ring over a ring R with multiplier s. For a general local ring R and a central element s in the Jacobson radical of R, necessary and sufficient conditions are obtained for K s(R) to be a strongly clean ring. For a commutative local ring R and an arbitrary element s in R, criteria are obtained for a single element of K s(R) to be strongly clean and, respectively, for the ring K s(R) to be strongly clean. Specializing to s =1 yields some known results. New families of strongly clean rings are presented. [Copyright &y& Elsevier]

Abstract: This paper extends Hua’s theorem on the geometry of rectangular matrices over a division ring to the case of Bezout domains. Let be integers , R an be two Bezout domains. Assume that is an adjacency preserving bijective map in both directions. Further, assume that is a local ring, or is an invertibility preserving map, or is an additive map. This paper obtains the algebraic formulas of . As applications, the ring semi-isomorphisms from to are characterized, and the group isomorphisms from to are discussed. [Copyright &y& Elsevier]

Abstract: Let be a commutative, local, and principal ideal ring with maximal ideal and residue class field . Suppose that every element of is square. Then the problem of classifying arbitrary symmetric matrices over by congruence naturally reduces, and is actually equivalent to, the problem of classifying invertible symmetric matrices over by congruence. [Copyright &y& Elsevier]

Let R be a commutative local ring, and M the maximal ideal of R. We prove that if |R/M|>3, then every matrix in SLnR (n⩾2) is a product of at most [n/2]+2 commutators of transvections and the length [res A/2]+2 cannot be reduced for n=2. [Copyright &y& Elsevier]