Invertible matrices over a class of semirings

We characterize the invertible matrices over a class of semirings such that the set of additively invertible elements is equal to the set of nilpotent elements. We achieve this by studying the liftings of the orthogonal sums of elements that are “almost idempotent” to those that are idempotent. Finally, we show an application of the obtained results to calculate the diameter of the commuting graph of the group of invertible matrices over the semirings in question.

The algebra generated by nilpotent elements in a matrix centralizer

For an arbitrary square matrix $S$, denote by $C(S)$ the centralizer of $S$, and by $C(S)_N$ the set of all nilpotent elements in $C(S)$. In this paper, we use the Weyr canonical form to study the subalgebra of $C(S)$ generated by $C(S)_N$. We determine conditions on $S$ such that $C(S)_N$ is a subalgebra of $C(S)$. We also determine conditions on $S$ such that the subalgebra generated by $C(S)_N$ is $C(S).$

Одно замечание о периодических кольцах

We obtain a new and non-trivial characterization of periodic rings (that are those rings $R$ for which, for each element $x$ in $R$, there exists two different integers $m$, $n$ strictly greater than $1$ with the property $x^m=x^n$) in terms of nilpotent elements which supplies recent results in this subject by Cui--Danchev published in (J. Algebra \& Appl., 2020) and by Abyzov--Tapkin published in (J. Algebra \& Appl., 2022). Concretely, we state and prove the slightly surprising fact that an arbitrary ring $R$ is periodic if, and only if, for every element~$x$ from $R$, there are integers $m>1$ and $n>1$ with $m\not= n$ such that the difference $x^m-x^n$ is a nilpotent.

Domination in the entire nilpotent element graph of a module over a commutative ring

Let R be a commutative ring with unity and M be a unitary R module. Let Nil(M) be the set of all nilpotent elements of M. The entire nilpotent element graph of M over R is an undirected graph E(G(M)) with vertex set as M and any two distinct vertices x and y are adjacent if and only if x + y ∈ Nil(M). In this paper we attempt to study the domination in the graph E(G(M)) and investigate the domination number as well as bondage number of E(G(M)) and its induced subgraphs N(G(M)) and Non(G(M)). Some domination parameters of E(G(M)) are also studied. It has been showed that E(G(M)) is excellent, domatically full and well covered under certain conditions.

Algebras associated with invariant means on the subnormal subgroups of an amenable group

Let G be an amenable group. We define and study an algebra ${\mathcal{A}}_{sn}(G)$, which is related to invariant means on the subnormal subgroups of G. For a just infinite amenable group G, we show that ${\mathcal{A}}_{sn}(G)$ is nilpotent if and only if G is not a branch group, and in the case that it is nilpotent we determine the index of nilpotence. We next study $\textrm{rad}\, \ell^1(G)^{**}$ for an amenable branch group G and show that it always contains nilpotent left ideals of arbitrarily large index, as well as non-nilpotent elements. This provides infinitely many finitely generated counterexamples to a question of Dales and Lau [4], first resolved by the author in [10], which asks whether we always have $(\textrm{rad}\, \ell^1(G)^{**})^{\Box 2} = \{0 \}$. We further study this question by showing that $(\textrm{rad}\, \ell^1(G)^{**})^{\Box 2} = \{0 \}$ imposes certain structural constraints on the group G.

Fuzzy Zorn’s lemma with applications

We introduced the fuzzy axioms of choice, fuzzy Zorn’s lemma and fuzzy well-ordering principle, which are the fuzzy versions of the axioms of choice, Zorn’s lemma and well-ordering principle, and discussed the relations among them. As an application of fuzzy Zorn’s lemma, we got the following results: (1) Every proper fuzzy ideal of a ring was contained in a maximal fuzzy ideal. (2) Every nonzero ring contained a fuzzy maximal ideal. (3) Introduced the notion of fuzzy nilpotent elements in a ring R, and proved that the intersection of all fuzzy prime ideals in a commutative ring R is the union of all fuzzy nilpotent elements in R. (4) Proposed the fuzzy version of Tychonoff Theorem and by use of fuzzy Zorn’s lemma, we proved the fuzzy Tychonoff Theorem.