Narrow Results

In this paper, we define several versions of the left extending condition for nearrings and rings not necessarily having a unity. We investigate the interconnections between the extending conditions and the Baer annihilating conditions in a right nearring. Our results culminate in a nearring version of the Chatters–Khuri theorem for rings. Examples are provided illustrating and delimiting our results.

A derivation of an associative ring R is an additive map satisfying T(xy) = T(x)y + xT(y) for all x, y in R. We study rings with a derivation T satisfying Herstein's condition [T(R), T(R)] = 0. (The commutator [u, v] is defined by: [u, v] = uv - vu.) This work studies the structure of the ideal I generated by T(R). We show that I³ is in the center of R, and we show that R has an ideal K which is contained in the kernel of T, K² = 0, and [T(R/K), R/K] generates a trivial ideal of R/K.

Let $S$ be a semigroup and $b,c\in S$. The concept of $(b,c)$-inverses was introduced by Drazin in 2012. It is well known that the Moore–Penrose inverse, the Drazin inverse, the Bott–Duffin inverse, the inverse along an element, the core inverse and dual core inverse are all special cases of the $(b,c)$-inverse. In this paper, a new relationship between the $(b,c)$-inverse and the Bott–Duffin $(e,f)$-inverse is established. The relations between the $(b,c)$-inverse of $paq$ and certain classes of generalized inverses of $pa$ and $aq$, and the $(b^{\prime },c^{\prime })$-inverse of $a$ are characterized for some $b^{\prime },c^{\prime }\in S$, where $p,a,q\in S$. Necessary and sufficient conditions for the existence of the $(B,C)$-inverse of a lower triangular matrix over an associative ring $R$ are also given, and its expression is derived, where $B,C$ are regular triangular matrices.

Given a complete modular meet-continuous lattice $A$, an inflator on $A$ is a monotone function $d:A\to A$ such that $a\le d(a)$ for all $a\in A$. If $I(A)$ is the set of all inflators on $A$, then $I(A)$ is a complete lattice. Motivated by preradical theory, we introduce two operators, the totalizer and the equalizer. We obtain some properties of these operators and see how they are related to the structure of the lattice $A$ and with the concept of dimension.

We classify modules and rings with some specific properties of their intersection graphs. In particular, we describe rings with infinite intersection graphs containing maximal left ideals of finite degree. This answers a question raised in [S. Akbari, R. Nikandish and J. Nikmehr, Some results on the intersection graphs of ideals of rings, J. Algebra Appl.12 (2013) 1250200]. We also generalize this result to modules, i.e. we get the structure theorem of modules for which their intersection graphs are infinite and contain maximal submodules of finite degree. Furthermore, we omit the assumption of maximality of submodules and still get a satisfactory characterization of such modules. In addition, we show that if the intersection graph of a module is infinite but its clique number is finite, then the clique and chromatic numbers of the graph coincide. This fact was known earlier only in some particular cases. It appears that such equality holds also in the complement graph.

Here, we continue to characterize a recently introduced notion, le-modules ${}_{R}M$ over a commutative ring $R$ with unity [A. K. Bhuniya and M. Kumbhakar, Uniqueness of primary decompositions in Laskerian le-modules, Acta Math. Hunga.158(1) (2019) 202–215]. This paper introduces and characterizes Zariski topology on the set Spec$(M)$ of all prime submodule elements of $M$. Thus, we extend many results on Zariski topology for modules over a ring to le-modules. The topological space Spec$(M)$ is connected if and only if R/Ann(M) contains no idempotents other than $\overline{0}$ and $\overline{1}$. Open sets in the Zariski topology for the quotient ring R/Ann(M) induces a base of quasi-compact open sets for the Zariski topology on Spec$(M)$. Every irreducible closed subset of Spec$(M)$ has a generic point. Besides, we prove a number of different equivalent characterizations for Spec$(M)$ to be spectral.

There is a special local ring $E$ of order $4,$ without identity for the multiplication, defined by $E=〈a,b|2a=2b=0,a^2=a,b^2=b,ab=a,ba=b〉.$ We study the algebraic structure of linear codes over that non-commutative local ring, in particular their residue and torsion codes. We introduce the notion of quasi self-dual codes over $E,$ and Type IV codes, that is quasi self-dual codes whose all codewords have even Hamming weight. We study the weight enumerators of these codes by means of invariant theory, and classify them in short lengths.

Let $R$ be a ring and let $n$ be an arbitrary but fixed positive integer. We characterize those rings $R$ whose elements $a$ satisfy at least one of the relations that $a^n+a$ or $a^n-a$ is a nilpotent whenever $n\in \mathbb{N}\setminus \{1\}$. This extends results from the same branch obtained by Danchev [A characterization of weakly J(n)-rings, J. Math. Appl. 41 (2018) 53–61], Koşan et al. [Rings with $x^n-x$ nilpotent, J. Algebra Appl. 19 (2020)] and Abyzov and Tapkin [On rings with $x^n-x$ nilpotent, J. Algebra Appl. 21 (2022)], respectively.