arXiv subjects
Publications and source records attributed to Ahmad Moussavi.
We systematically study those rings whose non-units are a sum of an idempotent and a nilpotent. Some crucial characteristic properties are completely described as well as some structural results for this class of rings are obtained. This work somewhat continues two publications on the subject due to Diesl (J. Algebra, 2013) and Karimi-Mansoub et al. (Contemp. Math., 2018).
Some variations of $π$-regular and nil clean rings were recently introduced in \cite{5,8,7}, respectively. In this paper, we examine the structure and relationships between these classes of rings. Specifically, we prove that $(m, n)$-regularly nil clean rings are left-right symmetric and also show that the inclusions ($D$-regularly nil clean) $\subseteq$ (regularly nil clean) $\subseteq$ ($(m,n)$-regularly nil clean) hold, as well as we answer Questions 1, 2 and 3 posed in \cite{8}. Moreover, some other analogous questions concerning the symmetric properties of certain classes of rings are treated as well by proving that centrally Utumi rings are always strongly $π$-regular.
Given a graded ample, Hausdorff groupoid $G$, and an involutive field $K$, we consider the Steinberg algebra $A_K(G)$. We obtain necessary and sufficient conditions on $G$ under which the annihilator of any graded ideal of $A_K(G)$ is generated by a homogeneous projection. This property is called graded quasi-Baer $\ast$. We use the Steinberg algebra model to characterize graded quasi-Baer $\ast$ Leavitt path algebras.
We consider in-depth and characterize in certain aspects those rings whose non-units are strongly nil-clean in the sense that they are a sum of commuting nilpotent and idempotent. In addition, we examine those rings in which the non-units are uniquely nil-clean in the sense that they are a sum of a nilpotent and an unique idempotent. In fact, we succeeded to prove that these two classes of rings can completely be characterized in terms of already well-studied and fully described sorts of rings.
We define and explore in-depth the notion of {\it UQ rings} by showing their important properties and by comparing their behavior with that of the well-known classes of UU rings and JU rings, respectively. Specifically, among the other established results, we prove that UQ rings are always Dedekind finite (often named directly finite) as well as that, for semipotent rings $R$, the following equivalence hold: $R/J(R)$ is UQ $\iff$ $R$ is UQ having the property that the set $QN(R)$ of quasinilpotent elements of $R$ coincides with the Jacobson radical $J(R)$ of $R$.
We continue the study in-depth of the so-called $n$-UU rings for any $n\geq 1$, that were defined by the first-named author in Toyama Math. J. (2017) as those rings $R$ for which $u^n-1$ is always a nilpotent for every unit $u\in R$. Specifically, for any $n\geq 2$, we prove that a ring is strongly $n$-nil-clean if, and only if, it is simultaneously strongly $π$-regular and an $(n-1)$-UU ring. This somewhat extends results due to Diesl in J. Algebra (2013), Abyzov in Sib. Math. J. (2019) and Cui-Danchev in J. Algebra Appl. (2020). Moreover, our results somewhat improves the ones obtained by Ko$ş$an et al. in Hacettepe J. Math. Stat. (2020).
We study those rings in which all invertible elements are weakly nil-clean calling them {\it UWNC rings}. This somewhat extends results due to Karimi-Mansoub et al. in Contemp. Math. (2018), where rings in which all invertible elements are nil-clean were considered abbreviating them as {\it UNC rings}. Specifically, our main achievements are that the triangular matrix ring ${\rm T}_n(R)$ over a ring $R$ is UWNC precisely when $R$ is UNC. Besides, the notions UWNC and UNC do coincide when $2 \in J(R)$. We also describe UWNC $2$-primal rings $R$ by proving that $R$ is a ring with $J(R) = {\rm Nil}(R)$ such that $U(R)=\pm 1+{\rm Nil}(R)$. In particular, the polynomial ring $R[x]$ over some arbitrary variable $x$ is UWNC exactly when $R$ is UWNC. Some other relevant assertions are proved in the present direction as well.
We define the class of {\it CUSC} rings, that are those rings whose clean elements are uniquely strongly clean. These rings are a common generalization of the so-called {\it USC} rings, introduced by Chen-Wang-Zhou in J. Pure \& Applied Algebra (2009), which are rings whose elements are uniquely strongly clean. These rings also generalize the so-called {\it CUC} rings, defined by Calugareanu-Zhou in Mediterranean J. Math. (2023), which are rings whose clean elements are uniquely clean. We establish that a ring is USC if, and only if, it is simultaneously CUSC and potent. Some other interesting relationships with CUC rings are obtained as well.
We define two types of rings, namely the so-called CSNC and NCUC that are those rings whose clean elements are strongly nil-clean, respectively, whose nil-clean elements are uniquely clean. Our results obtained in this paper somewhat expand these obtained by Calugareanu-Zhou in Mediterr. J. Math. (2023) and by Cui-Danchev-Jin in Publ. Math. Debrecen (2024), respectively.
Birkenmeier and Heider, in [2], say that a ring R is right cP-Baer if the right annihilator of a cyclic projective right R-module in R is generated by an idempotent. These rings are a generalization of the right p.q.-Baer and abelian rings. Generally, a formal power series ring over one indeterminate, wherein its base ring is right p.q.-Baer, is not necessarily right p.q.-Baer. However, according to [2], if the base ring is right cP-Baer then the power series ring over one indeterminate is right cP-Baer. Following [2], we investigate the transfer of the cP-Baer property between a ring R and various extensions (including skew polynomials, skew Laurent polynomials, skew power series, skew inverse Laurent series, and monoid rings). We also answer a question posed by Birkenmeier and Heider [2] and provide examples to illustrate the results.