Related papers: Matrices over a commutative ring as sums of three …
We will use commutators to provide decompositions of $3\times 3$ matrices as sums whose terms satisfy some polynomial identities, and we apply them to bounded linear operators and endomorphisms of free modules of infinite rank. In…
We present new characterizations of the rings in which every element is the sum of two idempotents and a nilpotent that commute, and the rings in which every element is the sum of two tripotents and a nilpotent that commute. We prove that…
In a recent article, we gave a full characterization of matrices that can be decomposed as a linear combination of two idempotents with prescribed coefficients. In this one, we use those results to improve on a recent theorem of V.…
Idempotent elements play a fundamental role in ring theory, as they encode significant information about the underlying algebraic structure. In this paper, we study idempotent matrices from two perspectives. First, we analyze the partially…
We study a ring containing a complete set of orthogonal idempotents as a generalized matrix ring via its Peirce decomposition. We focus on the case where some of the underlying bimodule homomorphisms are zero. Upper and lower triangular…
For any $n\ge 2$ and fixed $k\ge 1$, we give necessary and sufficient conditions for an arbitrary nonzero square matrix in the matrix ring $\mathbb{M}_n(\mathbb{F})$ to be written as a sum of an invertible matrix $U$ and a nilpotent matrix…
We study which matrices are sums of idempotents over a field of non-zero characteristic; in particular, we prove that any such matrix, provided it is large enough, is actually a sum of five idempotents, and even of four when the field is a…
In order to find a suitable expression of an arbitrary square matrix over an arbitrary finite commutative ring, we prove that every such a matrix is always representable as a sum of a potent matrix and a nilpotent matrix of order at most…
We investigate invertible matrices over finite additively idempotent semirings. The main result provides a criterion for the invertibility of such matrices. We also give a construction of the inverse matrix and a formula for the number of…
On the Waring's problems for matrices over a commutative ring, there are some trace conditions provided for matrices eligibly expressed as a sum of $k$-th powers with $k=2,3,4,5,6,7,8$ in several literatures. In this paper, we provide the…
In this paper, we use the idempotent decomposition to give an explicit isomorphism from an arbitrary semisimple Artinian ring to an external direct sum of finitely many full matrix rings over division rings.
We study several variants of decomposing a symmetric matrix into a sum of a low-rank positive semidefinite matrix and a diagonal matrix. Such decompositions have applications in factor analysis and they have been studied for many decades.…
An arbitrary Mueller matrix can be decomposed into a sum of up to four deterministic Mueller-Jones matrices, with strengths given by the eigenvalues of an associated Hermitian matrix. A geometrical representation of the eigenvalues in terms…
In this paper we consider the decomposition of positive semidefinite matrices as a sum of rank one matrices. We introduce and investigate the properties of various measures of optimality of such decompositions. For some classes of positive…
A ring R is Zhou nil-clean if every element in R is the sum of two tripotents and a nilpotent that commute. Let R be a Zhou nil-clean ring. If R is 2-primal (of bounded index), we prove that every square matrix over R is the sum of two…
A ring $R$ is (strongly) 2-nil-clean if every element in $R$ is the sum of two idempotents and a nilpotent (that commute). Fundamental properties of such rings are discussed. Let $R$ be a 2-primal ring. If $R$ is strongly 2-nil-clean, we…
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…
We characterize the idempotent stable range one $2\times 2$ matrices over commutative rings and in particular, the integral matrices with this property. Several special cases and examples complete the subject.
In this paper we deal with the problem of computing the sum of the $k$-th powers of all the elements of the matrix ring $\mathbb{M}_d(R)$ with $d>1$ and $R$ a finite commutative ring. We completely solve the problem in the case…
An element of a ring $R$ is called strongly $J^{\#}$-clean provided that it can be written as the sum of an idempotent and an element in $J^{\#}(R)$ that commute. We characterize, in this article, the strongly $J^{\#}$-cleanness of matrices…