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This article aims to solve positively Anderson-Badawi Conjecture of n-Absorbing and strongly n-absorbing ideals of commutative rings in the class of u-rings. The main result extends and recovers Anderson-Badawis related result on Prufer…

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…

This article introduces patterns of ideals of numerical semigroups, thereby unifying previous definitions of patterns of numerical semigroups. Several results of general interest are proved. More precisely, this article presents results on…

We describe first-degree prime ideals of biquadratic extensions in terms of first-degree prime ideals of two underlying quadratic fields. The identification of the prime divisors is given by numerical conditions involving their ideal norms.…

Following G.Szasz [2] a subsemigroup I of semigroup S is called an interior ideal if SIS \subset I. In this paper we explore the classes of regular semigroup and its different subclasses by their interior ideals. Furthermore, we introduce…

In the nonlinear field of multilinear operators and homogeneous polynomials between Banach spaces, we develop a technique, based on the transformation of vector-valued sequences, to create new examples of hyper-ideals of multilinear…

In this paper we describe the prime ideals of some important classes of skew PBW extensions, using the classical technique of extending and contracting ideals. Skew PBW extensions include as particular examples Weyl algebras, enveloping…

The purpose of this article is to define and examine graded almost prime ideals over a non-commutative graded ring, and consider some cases where all graded right ideals of a non-commutative graded ring are graded almost prime.

Let $R$ be a commutative ring with identity. In this paper, we introduce the concept of weakly $1$-absorbing prime ideals which is a generalization of weakly prime ideals. A proper ideal $I$ of $R$ is called weakly $1$-absorbing prime if…

Let $R$ be a commutative ring with. The purpose of this paper is to introduce and investigate cubes-difference factor absorbing ideals of R as a generalization of prime ideals.

We introduce the notion of Krull super-dimension of a super-commutative super-ring. This notion is used to describe regular super-rings and calculate Krull super-dimensions of completions of super-rings. Moreover, we use this notion to…

Let R be a commutative ring with identity. In this paper, we introduce the concept of (m, n)-closed ideals of R and (m, n)-von Neumann regular rings

Let (R,m) be a regular local ring with prime ideals p and q such that p+q is m-primary and dim(R/p)+dim(R/q)=dim(R). It has been conjectured by Kurano and Roberts that p^{(n)} \cap q \subseteq m^{n+1} for all positive integers n. We discuss…

We study zero divisors and minimal prime ideals in semirings of characteristic one. Thereafter we find a counterexample to the most obvious version of primary decomposition, but are able to establish a weaker version. Lastly, we study…

This article focuses on approximately prime rings and approximately prime ideals in proximal relator spaces, especially in descriptive proximity spaces. In particular, we define some binary operations, including the product of two…

Alan and al. defined and studied quasi-maximal ideals. We add a comprehensive characterization of these ideals, introducing submaximal ideals. The conductor of a finite minimal extension $R\subset S$ is quasi-maximal in $S$. This allows us…

In this paper, we introduce the concept of S-J-ideals in both commutative and noncommutative rings. For a commutative ring R and a multiplicatively closed subset S, we show that many properties of J-ideals apply to S-J-ideals and examine…

In this article, we define the concept of an $S$-$k$-irreducible ideal and $S$-$k$-maximal ideal in a commutative semiring. We also establish several results concerning $S$-$k$-primary ideals and prove the existence theorem and the…

We introduce extremely symmetric primes and provide some elementary properties of these.

This paper considers graded near-rings over a monoid G as a generalizations of the graded rings over groups, introduce certain innovative graded weakly prime ideals and graded almost prime ideals as a generalizations of graded prime ideals…