Related papers: Multiplicative lattices with absorbing factorizati…

Let $R$ be a commutative ring with non-zero identity and $M$ be a unitary $R$-module. The goal of this paper is to extend the concept of 1-absorbing primary ideals to 1-absorbing primary submodules. A proper submodule $N$ of $M$ is said to…

Let $G$ be a group, $R$ be a $G$-graded commutative ring with nonzero unity and $GI(R)$ be the set of all graded ideals of $R$. Suppose that $\phi:GI(R)\rightarrow GI(R)\cup\{\emptyset\}$ is a function. In this article, we introduce and…

The notion of $n$-normal residuated lattice, as a class of residuated lattices in which every prime filter contains at most $n$ minimal prime filters, is introduced and studied. Before that, the notion of $\omega$-filter is introduced and…

Let $G$ be a group with identity $e$ and $R$ be a $G$-graded commutative ring with nonzero unity $1$. In this article, we introduce the concept of graded strongly $1$-absorbing primary ideals. A proper graded ideal $P$ of $R$ is said to be…

In this paper, we introduce $\phi$-$\delta$-primary elements in a compactly generated multiplicative lattice $L$ and obtain its characterizations. We prove many of its properties and investigate the relations between these structures. By a…

In this study, we introduce the concept of "uniformly 2-absorbing primary ideals" of commutative rings, which imposes a certain boundedness condition on the usual notion of 2-absorbing primary ideals of commutative rings. Then we…

Let $R$ be a commutative ring with nonzero identity. Let $\mathcal{I}(R)$ be the set of all ideals of $R$ and let $\delta : \mathcal{I}(R)\longrightarrow \mathcal{I}(R)$ be a function. Then $\delta$ is called an expansion function of ideals…

Finite elements, which are well-known and studied in the framework of vector lattices, are investigated in $\ell$-algebras, preferably in $f$-algebras, and in product algebras. The additional structure of an associative multiplication leads…

The aim of this paper is to investigate further properties of $z$-elements in multiplicative lattices. We utilize $z$-closure operators to extend several properties of $z$-ideals to $z$-elements and introduce various distinguished…

We study the two dual notions of prime avoidance and prime absorbance. We generalize the classical prime avoidance lemma to radical ideals. A number of new criteria are provided for an abstract ring to be C.P. (every set of primes satisfies…

In this paper we study the concept of radical factorization in the context of abstract ideal theory in order to obtain a unified approach to the theory of factorization into radical ideals and elements in the literature of commutative…

Let G be a finite group acting on {1,...,n}. For any C*-algebra A, this defines an action of \alpha of G on A^{\otimes n}. We show that if A tensorially absorbs a UHF algebra of infinite type, the Jiang-Su algebra, or is approximately…

We prove that if $\Gamma$ is an icc irreducible lattice in a product of connected non-compact rank one simple Lie groups with finite center, then the II$_1$ factor $L(\Gamma)$ is prime. In particular, we deduce that the II$_1$ factors…

We define a notion of tracial $\mathcal{Z}$-absorption for simple not necessarily unital C*-algebras, study it systematically, and prove its permanence properties. This extends the notion defined by Hirshberg and Orovitz for unital…

In this paper, all rings are commutative with nonzero identity. Let M be an R-module. We introduce the concept of phi classical 1-absorbing prime submodules. A proper submodule N of M is a phi classical 1-absorbing prime submodule if…

Let $R$ be a commutative ring with identity, $S$ a multiplicative subset of $R$ and $I$ an ideal of $R$ disjoint from $S$. In this paper, we introduce the notion of an $S$-$n$-absorbing ideal which is a generalization of both the $S$-prime…

All rings are commutative with $1$ and $n$ is a positive integer. Let $\phi: J(R)\to J(R)\cup{\emptyset}$ be a function where $J(R)$ denotes the set of all ideals of $R$. We say that a proper ideal $I$ of $R$ is $\phi$-$n$-absorbing primary…

Let $G$ be a group with identity $e$, $R$ be a commutative $G$-graded ring with unity $1$ and $M$ be a $G$-graded unital $R$-module. In this article, we introduce the concept of graded $1$-absorbing prime submodule. A proper graded…

A distributive lattice $L$ with minimum element $0$ is called decomposable lattice if $a$ and $b$ are not comparable elements in $L$ there exist $\overline{a},\overline{b}\in L$ such that $a=\overline{a}\vee(a\wedge b),…

In this paper, we introduce a concept of $\mathfrak{X}$-element with respect to an $M$-closed set $\mathfrak{X}$ in multiplicative lattices and study properties of $\mathfrak{X}$-elements. For a particular $M$-closed subset $\mathfrak{X}$,…