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A CHARACTERISATION OF MATRIX RINGS.
- Source :
-
Bulletin of the Australian Mathematical Society . Feb2023, Vol. 107 Issue 1, p95-101. 7p. - Publication Year :
- 2023
-
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. [ABSTRACT FROM AUTHOR]
- Subjects :
- *MATRIX rings
Subjects
Details
- Language :
- English
- ISSN :
- 00049727
- Volume :
- 107
- Issue :
- 1
- Database :
- Academic Search Index
- Journal :
- Bulletin of the Australian Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 161234435
- Full Text :
- https://doi.org/10.1017/S0004972722000697