Related papers: A complete radical formula and 2-primal modules
Let R be a commutative ring with identity and M be an R-module. In this paper, we will introduce the concept of 2-irreducible (resp., strongly 2- irreducible) submodules of M as a generalization of irreducible (resp., strongly irreducible)…
We define topologically semiperfect (complete, separated, right linear) topological rings and characterize them by equivalent conditions. We show that the endomorphism ring of a module, endowed with the finite topology, is topologically…
Rigid monoidal 1-categories are ubiquitous throughout quantum algebra and low-dimensional topology. We study a generalization of this notion, namely rigid algebras in an arbitrary monoidal 2-category. Examples of rigid algebras include…
Principal affine open subsets in affine schemes are an important tool in the foundations of algebraic geometry. Given a commutative ring $R$, $\,R$-modules built from the rings of functions on principal affine open subschemes in…
The notion of clean rings and 2-good rings have many variations, and have been widely studied. We provide a few results about two new variations of these concepts and discuss the theory that ties these variations to objects and properties…
Let $G$ be a group with identity $e$. Let $R$ be a $G$-graded commutative ring and $M$ a graded $R$-module. In this paper, we introduce the concept of graded classical and graded strongly classical 2-absorbing second submodules of graded…
In this paper several characterizations of semi-compact modules are given. Among other results, we study rings whose semi-compact modules are injective. We introduce the property $\Sigma$-semi-compact for modules and we characterize the…
Let $R$ be a commutative noetherian ring. The $n$-semidualizing modules of $R$ are generalizations of its semidualizing modules. We will prove some basic properties of $n$-semidualizing modules. Our main result and example shows that the…
Let R be a commutative ring with identity and S a multiplicative subset of R. The aim of this paper is to study the class of commutative rings in which every S-flat module is flat (resp., projective). An R-module M is said to be S-flat if…
The fundamental theorem of symmetric polynomials over rings is a classical result which states that every unital commutative ring is fully elementary, i.e. we can express symmetric polynomials with elementary ones in a unique way. The…
We investigate homological properties of perfect algebras of prime characteristic. The principle is as follows: perfect algebras resolve the singularities. For example, we show any module over the ring of absolute integral closure has…
In this article, all rings are commutative with nonzero identity. Let $M$ be an $R$-module. A proper submodule $N$ of $M$ is called a classical prime submodule, if for each $m\in M$ and elements $a,b\in R$, $abm\in N$ implies that $am\in N$…
Purpose: To develop the algebraic foundation of finite commutative ternary $\Gamma$-semirings by identifying their intrinsic invariants, lattice organization, and radical behavior that generalize classical semiring and $\Gamma$-ring…
A ring $R$ is (strongly) 2-nil-clean if every element in $R$ is the sum of two idempotents and a nilpotent (that commute). Fundamental properties of such rings are discussed. Let $R$ be a 2-primal ring. If $R$ is strongly 2-nil-clean, we…
Inspired by the perspective of Reyes' noncomutative spectral theory, we attempt to develop noncommutative algebraic geometry by introducing ringed coalgebras, which can be thought of as a noncommutative generalization of schemes over a…
In this paper we study commutative rings $R$ whose prime ideals are direct sums of cyclic modules. In the case $R$ is a finite direct product of commutative local rings, the structure of such rings is completely described. In particular, it…
Let R be a commutative ring with identity and S be a multiplicatively closed subset of R. The aim of this paper is to introduce the notion of fully S-idempotent modules as a generalization of fully idempotent modules and investigate some…
Let $R$ be a commutative ring with identity. For an $R$-module $M$, the notion of strongly prime submodule of $M$ is defined. It is shown that this notion of prime submodule inherits most of the essential properties of the usual notion of…
Starting from the notion of totally reflexive modules, we survey the theory of Gorenstein homological dimensions for modules over commutative rings. The account includes the theory's connections with relative homological algebra and with…
In this paper, we study the classes of rings in which every proper (regular) ideal can be factored as an invertible ideal times a nonempty product of proper radical ideals. More precisely, we investigate the stability of these properties…