Search

We give a constructive sufficient condition for a matrix over a commutative ring to be von Neumann regular, and we show that it is also necessary over local rings. Specifically, we prove that a matrix A over a local commutative ring is von Neumann regular if and only if A has an invertible ρ (A) × ρ (A) -submatrix if and only if the determinantal rank ρ (A) and the McCoy rank of A coincide. We deduce consequences to (products of local) commutative rings, and we determine the number of von Neumann regular matrices over some finite rings of residue classes and group algebras. [ABSTRACT FROM AUTHOR]

Let R be a finite commutative ring with identity, n ∈ N and β a bilinear form on R n . In this work, we count the numbers of free submodules and totally isotropic free submodules of R n of rank s by using the lifting idea. We define the graph whose vertex set is the set of totally isotropic free submodules of R n of rank s called the generalized bilinear form graph. We study this graph when (R n , β) is a symplectic space and an orthogonal space. We can determine the degree of each vertex of these graphs. If R is a finite local ring, we show that these graphs are arc transitive and obtain their automorphism groups. We complete our work by proving that we can decompose the graphs over a finite commutative ring into the tensor products of graphs over finite local rings. [ABSTRACT FROM AUTHOR]

Orthogonal decomposition of the special linear Lie algebra over the complex numbers was studied in the early 1980s and attracted further attentions in the past decade due to its application in quantum information theory. In this paper, we study this decomposition problem of the special linear Lie algebra over a finite commutative ring with identity. [ABSTRACT FROM AUTHOR]

In this paper, we provide concrete constructions of idempotents to represent typical singular matrices over a given ring as a product of idempotents and apply these factorizations for proving our main results. We generalize works due to Laffey (Products of idempotent matrices.Linear Multilinear A.1983) and Rao (Products of idempotent matrices.Linear Algebra Appl.2009) to noncommutative setting and fill in the gaps in the original proof of Rao’s main theorems. We also consider singular matrices over Bézout domains as to when such a matrix is a product of idempotent matrices. [ABSTRACT FROM AUTHOR]