Related papers: Rings on quotient divisible abelian groups

Let $R$ be a commutative ring. If the nilpotent radical $Nil(R)$ of $R$ is a divided prime ideal, then $R$ is called a $\phi$-ring. In this paper, we first distinguish the classes of nonnil-coherent rings and $\phi$-coherent rings…

A pair of ideals $J\subseteq I\subseteq R$ has been called Aluffi torsion-free if the Aluffi algebra of $I/J$ is isomorphic with the corresponding Rees algebra. We give necessary and sufficient conditions for the Aluffi torsion-free…

In 1960, L\'{a}szl\'{o} Fuchs posed the problem of determining which groups $G$ are realizable as the group of units in some ring $R$. In \cite{chebolu2022fuchs}, we investigated the following variant of Fuchs' problem, for abelian groups:…

Let $RG$ be the group ring of an abelian group $G$ over a commutative ring $R$ with identity. An injection $\Phi$ from the subgroups of $G$ to the non-unit ideals of $RG$ is well-known. It is defined by $\Phi(N)=I(R,N)RG$ where $I(R,N)$ is…

We examine those matrix rings whose entries lie in periodic rings equipped with some additional properties. Specifically, we prove that the famous Diesl's question whether or not $R$ being nil-clean implies that $\mathbb{M}_n(R)$ is…

Quasi-socle ideals, that is the ideals $I$ of the form $I= Q : \mathfrak{m}^q$ in Gorenstein numerical semigroup rings over fields are explored, where $Q$ is a parameter ideal, and $\mathfrak{m}$ is the maximal ideal in the base local ring,…

An approach to identify the normal subgroups determined by ideals in free group rings with the help of the derived functors of non-additive functors is explored. A similar approach, i.e., via derived functors, for computing limits of…

We carry out a study of groups $G$ in which the index of any infinite subgroup is finite. We call them restricted-finite groups and characterize finitely generated not torsion restricted-finite groups. We show that every infinite…

It is proved that centrally essential rings, whose additive groups of finite rank are torsion-free groups of finite rank, are quasi-invariant but not necessarily invariant. Torsion-free Abelian groups of finite rank with centrally essential…

A ring R is called an E-ring if the canonical homomorphism from R to the endomorphism ring End(R_Z) of the additive group R_Z, taking any r in R to the endomorphism left multiplication by r turns out to be an isomorphism of rings. In this…

We develop the theory of rational ideals for arbitrary associative algebras R without assuming the standard finiteness conditions, noetherianness or the Goldie property. The Amitsur-Martindale ring of quotients replaces the classical ring…

Let $I ~(\ne A)$ be an ideal of a $d$-dimensional Noetherian local ring $A$ with $\operatorname{ht}_AI \ge 2$, containing a non-zerodivisor. The problem of when the ring $I:I=\operatorname{End}_AI$ is Gorenstein is studied, in connection…

In this paper, we study the structure of the rings of Fricke characters of free abelian groups. More precisely, we consider a descending filtration of ideals in the ring, and give a basis of its graded quotients. Then by using this fact, we…

A classical problem, raised by Fuchs in 1960, asks to classify the abelian groups which are groups of units of some rings. In this paper, we consider the case of finitely generated abelian groups, solving Fuchs' problem for such group with…

We study the groups of rational points of abelian varieties defined over a finite field $ \mathbb{F}_q$ whose endomorphism rings are commutative, or, equivalently, whose isogeny classes are determined by squarefree characteristic…

We consider rings whose one-sided ideals are close to automorphism-invariant modules. We study rings in which every (finitely generated) right ideal is automorphism invariant and rings in which every right ideal is a finite direct sum of…

This paper deals with well-known notion of $PF$-rings, that is, rings in which principal ideals are flat. We give a new characterization of $PF$-rings. Also, we provide a necessary and sufficient condition for $R\bowtie I$ (resp., $R/I$…

We observe that the class of left and right artinian left and right morphic rings agrees with the class of artinian principal ideal rings. For $R$ an artinian principal ideal ring and $G$ a group, we characterize when $RG$ is a principal…

A ring $A$ is called Hopfian if $A$ cannot be isomorphic to a proper homomorphic image $A/J$. $A$ is called Bassian if there cannot be an injection of $A$ into a proper homomorphic image $A/J$. We consider classes of Hopfian and Bassian…

In \cite[Problem 72]{Fuchs60} Fuchs posed the problem of characterizing the groups which are the groups of units of commutative rings. In the following years, some partial answers have been given to this question in particular cases. In a…