This article is cited in 1 paper

MATHEMATICS

Nilpotent, nil-good, and nil-clean formal matrices over residue class rings

A. M. Elfimova, Ts. D. Norbosambuev, M. V. Podkorytov

Tomsk State University, Tomsk, Russian Federation

Abstract: Let us recall some classes of rings. $A$ ring $R$ is said to be fc-nil-clean if each element can be written as a sum of a nilpotent and $k$ idempotents. $A$ ring $R$ is said to be fine if each non-zero element can be written as a sum of a unit and a nilpotent. $A$ ring $R$ is called nil-good if every element is a nilpotent or a sum of a nilpotent and a unit. And, finally, ring $R$ is called nil-good clean if every element is a sum of a nilpotent, an idempotent, and a unit. In this paper, we continue our work on additive problems in formal matrix rings over residue class rings. We have found necessary and sufficient conditions for the nilpotency of a formal matrix over residue class rings. After that we have shown that a ring of such matrices is $(p-1)$-nil-clean and nil-good clean. Also, answering the question posed in the previous article of the second co-author, we prove that a ring of formal matrices over residue rings is never nil-good, and, therefore, not fine.

Keywords: formal matrix ring, nilpotent formal matrix, nil-good ring, fine ring, nil-clean ring, nil-good clean ring, Morita context ring.

UDC: 512.552

MSC: 08A35, 15B99, 16S50

Received: 28.02.2024

DOI: 10.17223/19988621/91/3