Narrow Results

Let $R$ be a ring. We give the equivalent categories of the module categories over the polynomial ring $R[x]$ and the quotient ring $R[x]/(x^n)$. As applications, we describe explicitly some important classes of modules over $R[x]$ and $R[x]/(x^n)$ such as finitely generated modules, projective modules and injective modules.

This paper concerns ring properties which are induced from the structure of the powers of prime ideals. An ideal $I$ of a ring $R$ is called $n$-primary (respectively, $T$-primary) provided that $AB\subseteq I$ for ideals $A,B$ of $R$ implies that $(A+I)/I$ or $(B+I)/I$ is nil of index $n$ (respectively, $(A+I)/I$ or $(B+I)/I$ is nil) in $R/I$, where $n\ge 1$. It is proved that for a proper ideal $I$ of a principal ideal domain $R$, $I$ is $T$-primary if and only if $I$ is of the form $p^{k}R$ for some prime element $p$ and $k\ge 1$ if and only if $I$ is $2$-primary, through which we study the structure of matrices over principal ideal domains. We prove that for a $T$-primary ideal $I$ of a ring $R$, $R/I$ is prime when the Wedderburn radical of $R/I$ is zero. In addition we provide a method of constructing strictly descending chain of $n$-primary radicals from any domain, where the $n$-primary radical of a ring $R$ means the intersection of all the $n$-primary ideals of $R$.

Let $R$ be a commutative ring with identity, $X$ be an indeterminate over $R$, $R[X]$ be the polynomial ring over $R$, $R[[X]]$ be the power series ring over $R$, and $Q$ be a principal prime ideal of $R[X]$ with $(Q\cap R)[X]⊊Q$. It is well known that if $R$ is an integral domain, then $R{[X]}_Q$ is a DVR and $R[X]$ has infinitely many such principal prime ideals. In this paper, among other things, we show that (i) $R_{Q\cap R}$ is a field, (ii) $R{[X]}_Q$ is a DVR, but (iii) there is a ring $R$ such that $R[X]$ has no principal prime ideal. We also study the maximal ideals of $R[[X]]$ that are principal.

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.

Let $k$ be a field of arbitrary characteristic, $A$ be a domain, and K = frac(A). Then

• ⁽¹⁾All exponential maps of $k^{[3]}$ are rigid, and we give a necessary and sufficient condition for the triangularity of $\delta \in \mathrm{\text{EXP}}(k^{[3]})$.
• ⁽²⁾If $\delta \in \mathrm{\text{EXP}}(A^{[3]})$ such that $\mathrm{\text{rank}}(\delta )=\mathrm{\text{rank}}({\delta }_K)$, then $\delta$ is rigid and we give a necessary and sufficient condition for the triangularity of $\delta$.

When $k$ is of zero characteristic, $(1)$ is due to [D. Daigle, A necessary and sufficient condition for triangulability of derivations of $k[X,Y,Z]$, J. Pure Appl. Algebra113(3) (1996) 297–305] and $(2)$ is due to [M. K. Keshari and S. A. Lokhande, A note on rigidity and triangulability of a derivation, J. Commut. Algebra6(1) (2014) 95–100].

We investigate Gorenstein projective, injective, flat modules and dimensions over a polynomial ring using the equivalent category of the module category over the polynomial ring. Several estimates of Gorenstein homological dimensions of modules over the polynomial ring are obtained.

Monomial ordering is the process of putting single polynomial terms, or monomials, in a certain order depending on predefined principles. It facilitates the systematic grouping and comparison of monomials, as well as the division and factorization of polynomials and other algebraic operations. Let $R:=𝕂[x_{i,j}|i\in [c],j\in \mathbb{N}]$, be a polynomial ring over arbitrary field $𝕂$, where $c$ is any fixed natural number. In this paper, we have defined different orderings like lexicographical, degree lexicographical and reverse degree lexicographical ordering on polynomial ring. We have discussed the infinitely many Inc-compatible term ordering on polynomial ring $R$ with respect to two different weight vectors $W_c$ and $W_{c,n}^{\prime }.$

Antoine studied conditions which are connected to the question of Amitsur of whether or not a polynomial ring over a nil ring is nil, observing the structure of nilpotent elements in Armendariz rings and introducing the notion of nil-Armendariz rings. The class of nil-Armendariz rings contains Armendariz rings and NI rings. We continue the study of nil-Armendariz rings, concentrating on the structure of rings over which coefficients of nilpotent polynomials are nilpotent. In the procedure we introduce the notion of CN-rings that is a generalization of nil-Armendariz rings. We first construct a CN-ring but not nil-Armendariz. This may be a base on which Antoine's theory can be applied and elaborated. We investigate basic ring theoretic properties of CN-rings, and observe various kinds of CN-rings including ordinary ring extensions. It is shown that a ring R is CN if and only if R is nil-Armendariz if and only if R is Armendariz if and only if R is reduced when R is a von Neumann regular ring.

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.

The study of symmetric rings has important roles in ring theory and module theory. We investigate the structure of ring properties related to symmetric rings and introduce H-symmetric and π-symmetric as generalizations. We construct a non-symmetric reversible ring whose basic structure is infinite-dimensional, comparing with the finite-dimensional such rings of Anderson, Camillo and Marks. The structure of π-reversible rings (with or without identity) of minimal order is completely investigated. The properties of zero-dividing polynomials over IFP rings are studied more to show that polynomial rings over symmetric rings are π-symmetric. It is also proved that all conditions in relation with our arguments in this paper are equivalent for regular or locally finite rings.

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.

In this paper, we study the annihilating properties of ideals generated by coefficients of polynomials and power series which satisfy a structural equation. We first show that if $f(x)Rg(x)=0$ for polynomials $f(x)={\sum }_{i=0}^m{a}_i{x}^i,g(x)={\sum }_{j=0}^m{b}_j{x}^j$ over any ring $R$, then for any $i,j$, there exist positive integers $s(i,j)$ and $t(i,j)$ such that $a_{i_1}R{a}_{i_2}R\cdots a_{i_{s(i,j)}}R{b}_j=0$ and $a_{i}R{b}_{j_1}R{b}_{j_2}R\cdots R{b}_{j_{t(i,j)}}=0$, whenever $i_1,i_2,\dots ,i_{s(i,j)}\le i$ and $j_1,j_2,\dots ,j_{t(i,j)}\le j$. Next we prove that if $f(x)Rg(x)=0$ for power series $f(x)={\sum }_{i=0}^{\infty }{a}_i{x}^i,g(x)={\sum }_{j=0}^{\infty }{b}_j{x}^j$ over any ring $R$, then for any $i,j$, there exist positive integers $s(i,j)$ and $t(i,j)$ such that $a_{i_{s(i,j)}}R\cdots R{a}_{i_2}R{a}_{i_1}R{b}_{j_1}R{b}_{j_2}R\cdots R{b}_{j_{t(i,j)}}=0$ when ${\sum }_{p=1}^{s(i,j)}{i}_p+{\sum }_{q=1}^{t(i,j)}{j}_q<(s(i,j)+1)(t(i,j)+1)(i+1)(j+1)$ and $i_p\le i$, $j_q\le j$ for each $p,q$.

Let 𝕜 be a field of characteristic p > 0 and R be a subalgebra of 𝕜[X] = 𝕜[x₁, …, x_n]. Let J(R) be the ideal in 𝕜[X] defined by . It is shown that if it is a principal ideal then , where q = pⁿ(p - 1)/2.

Antoine studied the structure of the set of nilpotent elements in Armendariz rings and introduced the concept of nil-Armendariz property as a generalization. Hong et al. studied Armendariz property on skew polynomial rings and introduced the notion of an α-Armendariz ring, where α is a ring monomorphism. In this paper, we investigate the structure of the set of α-nilpotent elements in α-Armendariz rings and introduce an α-nil-Armendariz ring. We examine the set of -nilpotent elements in a skew polynomial ring R[x;α], where is the monomorphism induced by the monomorphism α of an α-Armendariz ring R. We prove that every polynomial with α-nilpotent coefficients in a ring R is -nilpotent when R is of bounded index of α-nilpotency, and moreover, R is shown to be α-nil-Armendariz in this situation. We also characterize the structure of the set of α-nilpotent elements in α-nil-Armendariz rings, and investigate the relations between α-(nil-)Armendariz property and other standard ring theoretic properties.

Let (R, 𝔪) be a Noetherian local ring and M a finitely generated R-module with dim M = d. Following Cuong and the first author [N. T. Cuong and L. T. Nhan, J. Algebra267 (2003) 156–177], M is called pseudo Cohen–Macaulay if for a system of parameters of M, where . In this paper, first we improve some known results on pseudo Cohen–Macaulay modules. Then we study the localizations of pseudo Cohen–Macaulay modules in order to introduce the pseudo Cohen–Macaulayness for the non-local case. Finally, we give characterizations for the formal power series ring and the polynomial ring being pseudo Cohen–Macaulay.

We in this note consider the reflexive ring property on nil ideals, introducing the concept of a nil-reflexive ring as a generalization of the reflexive ring property. We will call a ring $R$nil-reflexive if $IJ=0$ implies $JI=0$ for nil ideals $I,J$ of $R$. The polynomial and the power series rings over a right Noetherian ring (or an NI ring) $R$ are shown to be nil-reflexive if ${(aRb)}^2=0$ implies $aRb=0$ for all $a,b\in N(R)$. We further investigate the structure of nil-reflexive rings, related to various sorts of ring extensions which have roles in ring theory.

Mason introduced the reflexive property for ideals, and recently this concept was extended to many sorts of subsets in rings. In this note, we restrict the reflexivity to nilpotent elements, and a ring will be said to be RNP if it satisfies this restriction. The structure of RNP rings is studied in relation to the near concepts and ring extensions which have roles in ring theory.

In this note, we focus our attention on a new ring structure related to annihilators, and consider a ring property that contains many kinds of ring classes, introducing right ZAFS. This property is shown to be not left-right symmetric but left-right symmetric for left or right Artinian rings. The left (right) ZAFS property is shown to pass to Ore extensions with automorphisms. The left (respectively, right) ZAFS property is shown to pass also to classical left (respectively, right) quotient rings, yielding that semiprime right Goldie rings are ZAFS.

The usual commutative ideal theory was extended to ideals in noncommutative rings by Lambek, introducing the concept of symmetric. Camillo et al. naturally extended the study of symmetric ring property to the lattice of ideals, defining the new concept of an ideal-symmetric ring. This paper focuses on the symmetric ring property on nil ideals, as a generalization of an ideal-symmetric ring. A ring $R$ will be said to be right (respectively, left) nil-ideal-symmetric if $IJK=0$ implies $IKJ=0$ (respectively, $JIK=0$) for nil ideals $I,J,K$ of $R$. This concept generalizes both ideal-symmetric rings and weak nil-symmetric rings in which the symmetric ring property has been observed in some restricted situations. The structure of nil-ideal-symmetric rings is studied in relation to the near concepts and ring extensions which have roles in ring theory.

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.