Narrow Results

The concept of an I-matrix in the full 2 × 2 matrix ring M₂(R/I), where R is an arbitrary UFD and I is a nonzero ideal in R, is introduced. We obtain a concrete description of the centralizer of an I-matrix in M₂(R/I) as the sum of two subrings 𝒮₁ and 𝒮₂ of M₂(R/I), where 𝒮₁ is the image (under the natural epimorphism from M₂(R) to M₂(R/I)) of the centralizer in M₂(R) of a pre-image of , and the entries in 𝒮₂ are intersections of certain annihilators of elements arising from the entries of . It turns out that if R is a PID, then every matrix in M₂(R/I) is an I-matrix. However, this is not the case if R is a UFD in general. Moreover, for every factor ring R/I with zero divisors and every n ≥ 3, there is a matrix for which the mentioned concrete description is not valid.

The concept of reflexive property is introduced by Mason. This note concerns a ring-theoretic property of matrix rings over reflexive rings. We introduce the concept of weakly reflexive rings as a generalization of reflexive rings. From any ring, we can construct weakly reflexive rings but not reflexive, using its lower nilradical. We study various useful properties of such rings in relation with ideals in matrix rings, showing that this new property is Morita invariant. We also investigate the weakly reflexive property of several sorts of ring extensions which have roles in ring theory.

Let K be a 2-torsion free ring with identity and R_n(K, J) be the ring of all n × n matrices over K such that the entries on and above the main diagonal are elements of an ideal J of K. We describe all Jordan derivations of the matrix ring R_n(K, J) in this paper. The main result states that every Jordan derivation Δ of R_n(K, J) is of the form Δ = D + Ω, where D is a derivation of R_n(K, J) and Ω is an extremal Jordan derivation of R_n(K, J).

A ring is said to satisfy the strong 2-sum property if every element is a sum of two commuting units. In this note, we present some sufficient or necessary conditions for the matrix ring over a commutative local ring to have the strong 2-sum property.

A $\ast$-ring $R$ is called a nil $\ast$-clean ring if every element of $R$ is a sum of a projection and a nilpotent. Nil $\ast$-clean rings are the $\ast$-version of nil-clean rings introduced by Diesl. This paper is about the nil $\ast$-clean property of rings with emphasis on matrix rings. We show that a $\ast$-ring $R$ is nil $\ast$-clean if and only if $J(R)$ is nil and $R/J(R)$ is nil $\ast$-clean. For a 2-primal $\ast$-ring $R$, with the induced involution given by$(a_{ij}{)}^{\ast }=(a_{ij}^{\ast }{)}^T$, the nil $\ast$-clean property of ${\mathbb{M}}_n(R)$ is completely reduced to that of ${\mathbb{M}}_n({\mathbb{Z}}_2)$. Consequently, ${\mathbb{M}}_n(R)$ is not a nil $\ast$-clean ring for $n=3,4$, and ${\mathbb{M}}_2(R)$ is a nil $\ast$-clean ring if and only if $J(R)$ is nil, $R/J(R)$is a Boolean ring and $a^{\ast }-a\in J(R)$ for all $a\in R$.

Let $R$ be an associative unital ring and not necessarily commutative. We analyze conditions under which every $n\times n$ matrix $A$ over $R$ is expressible as a sum $A=E_1+\dots +E_s+N$ of (commuting) idempotent matrices $E_i$ and a nilpotent matrix $N$.