相关论文: On SAP-rings

Let $R$ be a commutative ring with identity and $S$ a multiplicative subset of $R$. In this paper, we introduce and study the notions of $S$-pure $S$-exact sequences and $S$-absolutely pure modules which extend the classical notions of pure…

We introduce a notion of pure-minimality for chain complexes of modules and show that it coincides with (homotopic) minimality in standard settings, while being a more useful notion for complexes of flat modules. As applications, we…

In this paper, we study rings having the property that every right ideal is automorphism-invariant. Such rings are called right $a$-rings. It is shown that (1) a right $a$-ring is a direct sum of a square-full semisimple artinian ring and a…

In this paper, we introduce the concept of N-pure ideal as a generalization of pure ideal. Using this concept, a new and interesting type of rings is presented, we call it mid ring. Also, we provide new characterizations for von Neumann…

Let $R$ be a ring and $S$ a multiplicative subset of $R$. An $R$-module $T$ is called $u$-$S$-torsion ($u$- always abbreviates uniformly) provided that $sT=0$ for some $s\in S$. The notion of $u$-$S$-exact sequences is also introduced from…

This paper studies similarities and differences between the classes of rings over which each simple module is injective and rings over which each simple module is $\Sigma$-injective. The rings in the former class are called $V$-rings and…

We construct a family of semiprimitive and non von Neumann regular rings satisfying that any right or left module is isomorphic to a quotient of its flat cover (in the sense of Enochs) by a small submodule. This answers in the negative a…

Different and distinct notions of regularity for modules exist in the literature. When these notions are restricted to commutative rings, they all coincide with the well-known von-Neumann regularity for rings. We give new characterizations…

Separative von Neumann regular rings exist in abundance. For example, all regular self-injective rings, unit regular rings, regular rings with a polynomial identity are separative. It remains open whether there exists a non-separative…

We use the concept of a regular object with respect to another object in an arbitrary category, defined in \cite{dntd}, in order to obtain the transfer of regularity in the sense of Zelmanowitz between the categories $R-$mod and $S-$mod,…

Let $R$ be a commutative ring with identity and $S$ a multiplicative subset of $R$. In this paper, we first introduce and study the notions of $s$-pure exact sequences and $s$-absolutely pure modules which extend the classical notions of…

A ring $R$ is called right SSP (SIP) if the sum (intersection) of any two direct summands of $R_{R}$ is also a direct summand. Left sides can be defined similarly. The following are equivalent: (1) $R$ is right SSP. (2) $R$ is right C3 and…

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…

Flat modules play an important role in the study of the category of modules over rings and in the characterization of some classes of rings. We study the e-flatness for semimodules introduced by the first author using his new notion of…

We introduce and investigate the so-called D-regularly nil clean rings by showing that these rings are, in fact, a non-trivial generalization of the classical von Neumann regular rings and of the strongly $\pi$-regular rings. Some other…

If $R$ is a regular and semiartinian ring, it is proved that the following conditions are equivalent: (1) $R$ is unit-regular, (2) every factor ring of $R$ is directly finite, (3) the abelian group $K_0(R)$ is free and admits a basis which…

The set of idempotents of a regular semigroup is given an abstract characterization as a regular biordered set in [2], and in [4] it is shown how a biordered set can be associated with a complemented modular lattice. Von Neumann has shown…

A ring is clean (resp. almost clean) if each of its elements is the sum of a unit (resp. regular element) and an idempotent. In this paper we define the analogous notion for *-rings: a *-ring is *-clean (resp. almost *-clean) if its every…

In this paper, we give some new characterizations of $u$-$S$-Noetherian rings and $u$-$S$-coherent rings in terms of uniform $S$-version of injective precovers, flat preenvelopes and absolutely pure modules, respectively. Moreover, we give…

In this article, we give a complete characterization of semigroup graded rings which are graded von Neumann regular. We also demonstrate our results by applying them to several classes of examples, including matrix rings and groupoid graded…