Research Papers

Weakly strongly $2$-nil-clean rings

P. Danchev^a, M. Doostalizadeh^b, A. Moussavi<sup>b

a</sup> Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria
^b Department of Mathematics, Tarbiat Modares University, 14115-111 Tehran Jalal AleAhmad Nasr, Iran

Abstract: This paper is devoted to the introduction and exploration in-depth of the notion of weakly strongly $2$-nil-clean rings as a common nontrivial generalization of both strongly $2$-nil-clean rings and strongly weakly nil-clean rings as defined and studied by Chen–Sheibani in the J. Algebra & Appl. (2017). Specifically, it is proved that any weakly strongly $2$-nil-clean ring is strongly $\pi$-regular and, concretely, that it decomposes as the direct product of a strongly $2$-nil-clean ring and a ring of type $\mathbb{Z}_{5^k}$ for some $k\geq 1$. The results somewhat expand and supply those obtained by Salim-Shuker in the Eur. J. Pure & Appl. Math. (2023).

Keywords: Weakly nil-clean element (ring), weakly strongly nil-clean element (ring), matrix rings.

Received: 06.09.2025

Language: English