Semiregular, semiperfect and semipotent matrix rings relative to an ideal

Abstract

This paper investigates relative ring theoretical properties in the context of formal triangular matrix rings. The first aim is to study the semiregularity of formal triangular matrix rings relative to an ideal. We prove that the formal triangular matrix ring $T$ is $T'$-semiregular if and only if $A$ is $I$-semiregular, $B$ is $K$-semiregular and $N=M$ for an ideal $T'=\bigl(\begin{smallmatrix} I & 0\\ N & K \end{smallmatrix}\bigr)$ of $T=\bigl(\begin{smallmatrix} A & 0\\ M & B \end{smallmatrix}\bigr).$ We also discuss the relative semiperfect formal triangular matrix rings in relation to the strong lifting property of ideals. Moreover, we have considered the behavior of relative semipotent and potent property of formal triangular matrix rings. Several examples are provided throughout the paper in order to highlight our results.

Keywords

• Formal triangular matrix ring
• strongly lifting ideal
• semiregular ring
• semiperfect ring
• semipotent ring

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