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]

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]

Let R be a commutative local ring whose maximal ideal is generated by a nilpotent element, and Mat (n , R) be the multiplicative monoid of the square matrices of order n over R. In this paper, we provide the construction of Green's ℋ -equivalence classes in the multiplicative monoid Mat (n , R). Then, we enumerate these classes in the special cases R = ℤ / p d ℤ and R = q [ x ] / 〈 x d 〉. [ABSTRACT FROM AUTHOR]

We consider monomial matrices over a commutative local principal ideal ring R of type M(t,k,n) = Φ (Ik 0 0 tIn−k), 0n − 1 and Ik is the identity k×k matrix. In this paper, we indicate a criterion of the hereditary irreducibility of M(t,k,n) in the case t [k⋅(n−k)n +1 ≠ 0 . [ABSTRACT FROM AUTHOR]

In this paper, we study a class of local rings that are not isomorphic to the class of two local rings we have been working on before. This class is also an isomorphism to a special matrix algebra. We construct a projective coordinate space over a module defined on this special algebra class. Specifically, in a 3-dimensional projective coordinate space, the incidence matrix for a line that combines the certain two points and also all points of a line given with the incidence matrix are found by the help of Maple programme. [ABSTRACT FROM AUTHOR]

This paper is dealed with a special local ring A and modules over A. Some properties of modules, that are constructed over the real plural algebra, are investigated. Moreover a module is constructed over the linear algebra of matrix Mmm(R) and one of its basis is found. [ABSTRACT FROM AUTHOR]

A ring R is called pseudopolar if for every a ∈ R there exists p2 = p ∈ R such that p ∈ 2(a), a + p ∈ U(R) and akp ∈ J(R) for some positive integer k. Pseudopolar rings are closely related to strongly π-regular rings, uniquely strongly clean rings, semiregular rings and strongly π-rad clean rings. In this paper, we completely characterize the local rings R for which M2(R) is pseudopolar. [ABSTRACT FROM AUTHOR]

LetRbe a ring with identity,X(R) the set of all nonzero non-units ofRandG(R) the group of all units ofR. By considering left and right regular actions ofG(R) onX(R), the following are investigated: (1) For a local ringRsuch thatX(R) is a union ofndistinct orbits under the left (or right) regular action ofG(R) onX(R), ifJn ≠ 0 = Jn+1whereJis the Jacobson radical ofR, then the set of all the distinct ideals ofRis exactly {R,J,J2,…,Jn, 0}, and each orbit under the left regular action is equal to the one under the right regular action. (2) Such a ringRis left (and right) duo ring. (3) For the full matrix ringSofn × nmatrices over a commutative ringR, the number of orbits under left regular action ofG(S) onX(S) is equal to the number of orbits under right regular action ofG(S) onX(S); the result also holds for the ring ofn × nupper triangular matrices overR. [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]

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]

An element of a ring is called strongly J-clean provided that it can be written as the sum of an idempotent and an element in its Jacobson radical that commute. We investigate, in this article, a single strongly J-clean 2 × 2 matrix over a noncommutative local ring. The criteria on strong J-cleanness of 2 × 2 matrices in terms of a quadratic equation are given. These extend the corresponding results in [8, Theorems 2.7 and 3.2], [9, Theorem 2.6], and [11, Theorem 7]. [ABSTRACT FROM PUBLISHER]

A ring R is quasipolar if for any a ∈ R, there exists p 2 = p ∈ R such that , a + p ∈ U(R) and ap ∈ R qnil . In this article, we investigate conditions on a local ring R that imply every n × n upper triangular matrix ring over R is quasipolar. It is shown that this is the case for commutative local rings, as well as for a host of other classes of local rings. [ABSTRACT FROM PUBLISHER]

An element of a ring is called strongly J-clean provided that it can be written as the sum of an idempotent and an element in its Jacobson radical that commute. A ring is strongly J-clean in case each of its elements is strongly J-clean. We investigate, in this article, strongly J-clean rings and ultimately deduce strong J-cleanness of Tn(R) for a large class of local rings R. Further, we prove that the ring of all 2 × 2 matrices over commutative local rings is not strongly J-clean. For local rings, we get criteria on strong J-cleanness of 2 × 2 matrices in terms of similarity of matrices. The strong J-cleanness of a 2 × 2 matrix over commutative local rings is completely characterized by means of a quadratic equation. [ABSTRACT FROM AUTHOR]

Let R be an associative ring with identity. An element a ∈ R is called strongly clean if a = e + u with e2 = e ∈ R, u a unit of R, and eu = ue. A ring R is called strongly clean if every element of R is strongly clean. Strongly clean rings were introduced by Nicholson [7]. It is unknown yet when a matrix ring over a strongly clean ring is strongly clean. Several articles discussed this topic when R is local or strongly π-regular. In this note, necessary conditions for the matrix ring n(R) (n > 1) over an arbitrary ring R to be strongly clean are given, and the strongly clean property of 2(RC2) over the group ring RC2 with R local is obtained. [ABSTRACT FROM AUTHOR]

In this article similarity classes of three by three matrices over a local principal ideal commutative ring are analyzed. When the residue field is finite, a generating function for the number of similarity classes for all finite quotients of the ring is computed explicitly. [ABSTRACT FROM AUTHOR]

Abstract: An element of a ring is called strongly clean if it can be written as the sum of a unit and an idempotent that commute. A ring is called strongly clean if each of its elements is strongly clean. In this paper, we investigate conditions on a local ring R that imply that is a strongly clean ring. It is shown that this is the case for commutative local rings R, as well as for a host of other classes of local rings. An example of a local ring A for which is not strongly clean is also given. [Copyright &y& Elsevier]