A CHARACTERISATION OF MATRIX RINGS. Issue 1 (8th February 2023)

• Record Type: Journal Article
• Authors: GOYAL, DIMPLE RANI; KHURANA, DINESH
• Is Part Of: Bulletin of the Australian Mathematical Society. Volume 107:Issue 1(2023)
• Page Start: 95
• Page End: 101
• Publication Date: 2023-02-08
• Subjects: 16U99 -- 16S50; matrix ring -- nilpotent element -- unit -- idempotent -- annihilator; Mathematics -- Societies, etc; Mathematics -- Periodicals; 510.5
• DOI: 10.1017/S0004972722000697
• Languages: English
• ISSNs: 0004-9727

Abstract

Abstract: We prove that a ring R is an $n \times n$ matrix ring (that is, $R \cong \mathbb {M}_n(S)$ for some ring S ) if and only if there exists a (von Neumann) regular element x in R such that $l_R(x) = R{x^{n-1}}$ . As applications, we prove some new results, strengthen some known results and provide easier proofs of other results. For instance, we prove that if a ring R has elements x and y such that $x^n = 0$, $Rx+Ry = R$ and $Ry \cap l_{R}(x^{n-1}) = 0$, then R is an $n \times n$ matrix ring. This improves a result of Fuchs ['A characterisation result for matrix rings', Bull. Aust. Math. Soc. 43 (1991), 265–267] where it is proved assuming further that the element y is nilpotent of index two and $x+y$ is a unit. For an ideal I of a ring R, we prove that the ring $(\begin {smallmatrix} R & I \\ R & R \end {smallmatrix})$ is a $2 \times 2$ matrix ring if and only if $R/I$ is so.