Related papers: On Dependent Elements of Semiprime Rings
Inner relations are derived between partial augmentations of certain elements (units or idempotents) in group rings.
We investigate bicomplex analogues of fundamental notions from classical algebraic number theory. In particular, we show that the primitive element theorem admits a natural generalization to bicomplex extensions, giving rise to two distinct…
In this paper, we introduce the concept of n-semiprimary ideals, n-powerful ideals, and n-powerful semiprimary ideals of commutative rings. We study these concepts and relate them to several generalizations of pseudo-valuation domains.
Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring over a field and $A$ a standard graded $S$-algebra. In terms of the Gr\"obner basis of the defining ideal $J$ of $A$ we give a condition, called the x-condition, which implies that all graded…
Let $R$ be a ring and $P$ a prime ideal of $R.$ In this paper, we establish some commutativity criteria for the factor ring $R/P$ in terms of derivations of $R$ satisfying some algebraic identities involving a new kind of involution in…
The implementation of a spin qubit in a quantum ring occupied by one or a few electrons is proposed. Quantum bit involves the Zeeman sublevels of the highest occupied orbital. Such a qubit can be initialized, addressed, manipulated, read…
In this note, we revisit certain results from a previous work of the second author concerning $z$-ideals of semirings. We demonstrate that the assumption of the semiring being a bzi-semiring can be dispensed with. Additionally, we propose a…
Let k be an algebraically closed field of characteristic zero and let B be a finitely generated k-domain. We study semisimple derivations on B, with special emphasis on those whose eigenvalues are integers. For such derivations, after…
We continue the analysis of prime and semiprime operations over one-dimensional domains started in \cite{Va}. We first show that there are no bounded semiprime operations on the set of fractional ideals of a one-dimensional domain. We then…
We explicitly describe the divisor class groups and semidualizing modules for ladder determinantal rings with coefficients in an arbitrary normal domain for arbitrary ladders, not necessarily connected, and all sizes of minors.
In this paper the question of which semigroups are realizable as the semigroup of values attained on a Noetherian local ring which is dominated by a valuation is considered. We give some striking examples, indicating that there may be no…
Let $R$ be a noncommutative ring, and let $S$ be an $m$-system of $R$. In this paper, we give more results on the concept of almost prime (right) ideals, that were introduced by the first two authors, especially in (right) $S$-unital rings,…
Let R be a commutative ring with identity and N(R) be the set of all nilpotent elements of R. The aim of this paper is to introduce and study the notion of nil-prime ideals as a generalization of prime ideals. We say that a proper ideal P…
In this paper, we introduce and study V- and CI-semirings---semirings all of whose simple and cyclic, respectively, semimodules are injective. We describe V-semirings for some classes of semirings and establish some fundamental properties…
Goldie's Theorem implies that a semiprime left Goldie ring is embeddable into a semisimple Artinian ring. On the other hand, there are domains that are not embeddable into division rings. A criterion for a semiprime ring being embeddable…
We define and explore the class of rings $R$ for which each element in $R$ is a sum of a tripotent element from $R$ and an element from the subring $\Delta(R)$ of $R$ which commute each other. Succeeding to obtain a complete description of…
A semiring is an algebraic structure similar to a ring, but without the requirement that each element must have an additive inverse. A po-semiring is a semiring equipped with a compatible bounded partial order. In this paper, properties of…
In this paper, we give some generalizations the concept of element order and we study some of the properties of these generalized order. In particular, with using this generalization we derive two solvability criteria.
In this paper we give an affirmative answer to two conjectures on generalized $(m,n)$-Jordan derivations and generalized $(m,n)$-Jordan centralizers raised in [S. Ali and A. Fo\v{s}ner, \textit{On Generalized $(m, n)$-Derivations and…
A ring $R$ with center $C$ is said to be centrally essential if the module $R_C$ is an essential extension of the module $C_C$. In this paper, we study properties of ideals of centrally essential rings, centrally essential quaternion…