Narrow Results

According to Jacobson, a right ideal of a ring $R$ is bounded if it contains a nonzero ideal of $R$. Faith called a ring strongly right bounded if every nonzero right ideal is bounded. We examine some annihilator conditions for skew inverse Laurent series ring $R((x^{-1};\sigma ,\delta ))$, accomplished with relationships among certain classical sorts of rings such as rings with bounded annihilators and rings with property ($A$).

We explore the concept of the $S$-Krull dimension in commutative rings, extending classical notions of Krull dimension by incorporating multiplicative subsets $S$ of a ring $R$. We provide characterizations of $S$-prime, $S$-maximal, and $S$-minimal $S$-prime ideals, establishing foundational results crucial for understanding the $S$-Krull dimension. The paper also delves into properties of $S$-principal ideal $S$-domains, $S$-Artinian rings, and u-$S$-von Neumann regular rings, introducing an $S$-variant of the Division Algorithm. In the final section, the concept of $S$-height for $S$-prime ideals is defined, and the $S$-Krull dimension is analyzed through various characterizations and examples, with a particular focus on polynomial rings. We conclude with insights into the $S$-height of larger $S$-prime ideals of $R[X]$ and their intersections with $R$, contributing to a deeper understanding of the $S$-Krull dimension.

In this paper, we study zero divisors in the Hurwitz series rings and the Hurwitz polynomial rings over general non-commutative rings. We first construct Armendariz rings that are not Armendariz of the Hurwitz series type and find various properties of (the Hurwitz series) Armendariz rings. We show that for a semiprime Armendariz of the Hurwitz series type (so reduced) ring $R$ with $a.c.c.$ on annihilator ideals, HR (the Hurwitz series ring with coefficients over $R$) has finitely many minimal prime ideals, say $B_1,\dots ,B_m$ such that $B_1\cdot \dots \cdot B_m=0$ and $B_i=H{A}_i$ for some minimal prime ideal $A_i$ of $R$ for all $i$, where $A_1,\dots ,A_m$ are all minimal prime ideals of $R$. Additionally, we construct various types of (the Hurwitz series) Armendariz rings and demonstrate that the polynomial ring extension preserves the Armendarizness of the Hurwitz series as the Armendarizness.

We continue the study of nil-Armendariz rings, initiated by Antoine, and Armendariz rings. We first examine a kind of ring coproduct constructed by Antoine for which the Armendariz, nil-Armendariz, and weak Armendariz properties are equivalent. Such a ring has an important role in the study of Armendariz ring property and near-related ring properties. We next prove an Antoine's result in relation with the ring coproduct by means of a simpler direct method. In the proof we can observe the concrete shapes of coefficients of zero-dividing polynomials. We next observe the structure of nil-Armendariz rings via the upper nilradicals. It is also shown that a ring R is Armendariz if and only if R is nil-Armendariz if and only if R is weak Armendariz, when R is a von Neumann regular ring.

This paper concerns several ring theoretic properties related to matrices and polynomials. The basic properties of $\pi$-reversible and power-Armendariz are studied. We provide a method by which one can always construct a power-Armendariz ring but neither symmetric nor Armendariz from given any symmetric ring. We investigate next various interesting relations among ring theoretic properties containing $\pi$-reversibility and power-Armendariz condition.

For a ring derivation δ, we introduce and investigate a generalization of reduced rings and Armendariz rings which we call a δ-Armendariz ring. Various classes of δ-Armendariz rings is provided and a number of properties of this generalization are established. Radicals and minimal prime ideals of the differential polynomial ring R[x; δ], in terms of those of a δ-Armendariz R, is determined. We prove that several properties transfer between R and the differential polynomial ring R[x; δ], in case R is δ-Armendariz.

We consider the ring R[x]/(xⁿ⁺¹), where R is a ring, R[x] is the ring of polynomials in an indeterminant x, (xⁿ⁺¹) is the ideal of R[x] generated by xⁿ⁺¹ and n is a positive integer. The aim of this paper is to show that regularity or strong regularity of a ring R is necessary and sufficient condition under which the ring R[x]/(xⁿ⁺¹) is an example of a ring which belongs to some important classes of rings. In this context, we discuss distributive rings, Bézout rings, Gaussian rings, quasi-morphic rings, semihereditary rings, and rings which have weak dimension less than or equal to one.

Let $R$ be a ring with compatible endomorphisms ${\alpha }_1,\dots ,{\alpha }_n$. We investigate the quasi-Armendariz, Armendariz, McCoy, semicommutative properties and Property (A) of the ring $S_n(R)$. Our results yield more examples of quasi-Armendariz rings, Armendariz rings, McCoy rings, semicommutative rings and rings with Property (A).

The purpose of this paper is to provide useful connections between units and zero divisors, by investigating the structure of a class of rings in which Köthe’s conjecture (i.e. the sum of two nil left ideals is nil) holds. We introduce the concept of unit-IFP for the purpose, in relation with the inserting property of units at zero products. We first study the relation between unit-IFP rings and related ring properties in a kind of matrix rings which has roles in noncommutative ring theory. The Jacobson radical of the polynomial ring over a unit-IFP ring is shown to be nil. We also provide equivalent conditions to the commutativity via the unit-IFP of such matrix rings. We construct examples and counterexamples which are necessary to the naturally raised questions.

In this paper, we introduce the concept of rings with right (left) weak property (A) which is a generalization of rings with right (left) property (A). A ring $R$ has right (left) weak property (A) if every finitely generated two-sided ideal $I$ of $R$ with $I\subseteq Z_l(R)(Z_r(R))$, there exists nonzero $u\in R$$(v\in R)$ such that $u\in Nr\cdot {\mathrm{\text{ann}}}_R(I)$$(v\in Nl\cdot {\mathrm{\text{ann}}}_R(I))$. Further, we study various extensions of rings with weak property (A) including matrix rings, polynomial rings and Ore extensions.

In this paper, we introduce the notion of an almost Armendariz ring, which is a generalization of an Armendariz ring, and discuss some of its properties. It has been found that every almost Armendariz ring is weak Armendariz but the converse is not true. We prove that a ring R is almost Armendariz if and only if R[x] is almost Armendariz. It is also shown that if R/I is an almost Armendariz ring and I is a semicommutative ideal, then R is an almost Armendariz ring. Moreover, the class of minimal non-commutative almost Armendariz rings is completely determined, up to isomorphism (minimal means having smallest cardinality).

In this paper, we tell about some results and open problems concerning Armendariz rings.

The focus of this paper is on a ring construction H_n(R; σ) based on a given ring R and a Hochschild 2-cocycle σ. This construction is a unified generalization of the ring R[x]/(xⁿ⁺¹) and the Hochschild extension H_σ(R, R). Here we discuss when the ring H_n(R; σ) is reversible, symmetric, Armendariz, abelian and uniquely clean, respectively. Several known results of R[x]/(xⁿ⁺¹) and H_σ(R, R) are extended to H_n(R; σ), and new examples of reversible, symmetric and Armendariz rings are given.