Semiregular, semiperfect and semipotent matrix rings relative to an ideal

Year 2024, Volume: 73 Issue: 1, 211 - 221, 16.03.2024

Meltem Altun Özarslan

https://doi.org/10.31801/cfsuasmas.1307158

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

References

• Altun Özarslan, M, Lifting properties of formal triangular matrix rings, RACSAM, 115(3) (2021), Paper No. 104, 13 pp. https://doi.org/10.1007/s13398-021-01044-0
• Berberian, S. K., The center of a corner of a ring, J. Algebra, 71(2) (1981), 515-523. https://doi.org/10.1016/0021-8693(81)90191-5
• Goodearl, K. R., Ring Theory. Nonsingular Rings and Modules, Monographs and Textbooks in Pure and Appl. Math., 33, Marcel Dekker, New York, 1976.
• Haghany, A, Varadarajan, K., Study of formal triangular matrix rings, Comm. Algebra, 27 (1999), 5507-5525. https://doi.org/10.1080/00927879908826770
• Herstein, I. N., A counter example in Noetherian rings, Proc. Nat. Acad. Sci. U.S.A., 54 (1965), 1036-1037. https://doi.org/10.1073/pnas.54.4.1036
• Hong, C. Y., Kim, N. K., Lee, Y., Exchange rings and their extensions, J. Pure Appl. Algebra, 179 (2003), 117-126. https://doi.org/10.1016/S0022-4049(02)00299-2
• Nicholson, W. K., Lifting idempotents and exchange rings, Trans. Amer. Math. Soc., 229 (1977), 269-278. https://doi.org/10.1090/S0002-9947-1977-0439876-2
• Nicholson, W. K., Yousif, M.F., Weakly continuous and C2-rings, Comm. Algebra, 29 (2001), 2429-2446. https://doi.org/10.1081/AGB-100002399
• Nicholson, W. K., Zhou, Y, Strong lifting, J. Algebra, 285(2) (2005), 795-818. https://doi.org/10.1016/j.jalgebra.2004.11.019
• Yousif, M. F., Zhou, Y, Semiregular, semiperfect and perfect rings relative to an ideal, Rocky Mountain J. Math., 32 (2002), 1651-1671. https://doi.org/10.1216/rmjm/1181070046