Narrow Results

In this survey, we study the defining ideals $𝔭$ of the space monomial curves $(t^a,t^b,t^c)$ for primitive Pythagorean triples (PPT) $a,b,c$. We give a minimal system of generators of $𝔭$. Hence, we provide the formula for Frobenius number of a PPT. Additionally, we show that over every field of arbitrary characteristic, there are distinct primitive Pythagorean triples such that their symbolic Rees algebras are noetherian.

Over a commutative ring the prime spectrum, as a geometrical object, reflects many properties of the ring. This geometrical object can be modified by the action of multiplicative closed subsets, or more in general, by hereditary torsion theories. For a multiplicatively closed subset $S$ of a commutative ring $A$, in the literature there were several $S$-noetherian spectrum properties. In this paper, for any commutative ring $A$, we introduce generalizations of them using a hereditary torsion theory $\sigma$ instead of a multiplicative closed subset $S\subseteq A$. Two different constructions of noetherian spectra on a ring, associated to a hereditary torsion theory, are considered in such a way that they are characterized by prime ideals and preserved by polynomial ring constructions.

A module $M$ is called pseudo semi-projective if, for all endomorphisms $\alpha ,\beta$ of $M$ with $\mathrm{\text{Im}}(\alpha )=\mathrm{\text{Im}}(\beta )$, then $\alpha \mathrm{\text{End}}(M)=\beta \mathrm{\text{End}}(M)$. In this paper, we study submodules of pseudo semi-projective modules. It is shown that if every (finitely generated) submodule of a semiprojective right $R$-module is pseudo semi-projective, then every factor ring of $R$ is right (semi-)hereditary. Moreover, we show that if $R$ is left perfect and finitely generated submodules of pseudo semi-projective right $R$-modules are pseudo semi-projective, then $R$ has a decomposition of abelian groups $R_{\mathbb{Z}}=S\oplus J(R)$, where $S$ is a semisimple subring of $R$ containing 1.

Let ${𝒜}={(A_n)}_{n\ge 0}$ be an increasing sequence of rings and $A={\bigcup }_{n\ge 0}{A}_n$. It is shown that the ring $H{𝒜}$ (respectively, $h{𝒜}$) is Noetherian if and only if (i) the ring $A_0$ is Noetherian, (ii) the sequence ${𝒜}$ is stationary, (iii) for each $n\ge 1$, the $A_0$-module $A_n$ is finitely generated and (iv) $\mathbb{Q}\subseteq A_0$. Also, we show that if $A\subseteq B$ is a ring extension and $I$ an ideal of $B$, the following assertions hold: (1) if the ring $H(A,I)$ (respectively, $h(A,I)$) is Noetherian, then (i) the ring $A$ is Noetherian, (ii) the $A$-module $I$ is finitely generated, (iii) the ideal $I$ of $B$ is idempotent and (iv) $\mathrm{\text{charact}}(A)=0$. Conversely, (2) if (i) the ring $A$ is Noetherian, (ii) the $A$-module $I$ is finitely generated, (iii) the ideal $I$ of $B$ is idempotent and (iv) $\mathbb{Q}\subseteq A$, then the ring $H(A,I)$ (respectively, $h(A,I)$) is Noetherian.

In previous work, the second author introduced a topology, for spaces of irreducible representations, that reduces to the classical Zariski topology over commutative rings but provides a proper refinement in various noncommutative settings. In this paper, a concise and elementary description of this refined Zariski topology is presented, under certain hypotheses, for the space of simple left modules over a ring R. Namely, if R is left noetherian (or satisfies the ascending chain condition for semiprimitive ideals), and if R is either a countable dimensional algebra (over a field) or a ring whose (Gabriel-Rentschler) Krull dimension is a countable ordinal, then each closed set of the refined Zariski topology is the union of a finite set with a Zariski closed set. The approach requires certain auxiliary results guaranteeing embeddings of factor rings into direct products of simple modules. Analysis of these embeddings mimics earlier work of the first author and Zimmermann-Huisgen on products of torsion modules.

In a series of recent papers, Beidar, Jain and Srivastava studied the question as to when a ring R with the property that essential extensions of semi-simple right R-modules are direct sums of quasi-injectives is right Noetherian. Beidar and Jain proved that it is, when R is commutative or right q.f.d. In this note we extend their results proving the following: A ring R with this property is right Noetherian iff for some n ∈ ℕ, R/soc_n(R_R) has ascending chain condition on essential non-two-sided right ideals (in particular, when R/soc_n(R_R) is right q.f.d. or commutative). Also shown is the following: A ring is a right Noetherian right V-ring iff modules with essential socle are quasi-continuous/quasi-injective.

We establish several characterizations of maximal non-Prüfer and maximal non-integrally closed subrings of a field. Special attention is given when finiteness conditions are satisfied. Several numerical characterizations are then obtained. The paper is concluded with examples that exhibit the obtained results.

Carl Faith (2003) introduced and investigated an interesting class of rings over which every cyclic right module has Σ-injective injective hull (abbr., right CSI-rings). Inspired by this we investigate rings over which every cyclic right R-module has a Σ-extending injective hull. We call such rings right CSE-rings and show that the class of right CSE-rings and that of right CSI-rings coincide. We also use other hulls of cyclic modules to define other classes of rings, and investigate their structure. We prove, among others, that a ring R is right QI if and only if the quasi-injective hull of each cyclic right module is Σ-injective.

We study classes of modules over a commutative ring which allow to do homological algebra relative to such a class. We classify those classes consisting of injective modules by certain subsets of ideals. When the ring is Noetherian the subsets are precisely the generization closed subsets of the spectrum of the ring.

Let R be an associative ring with identity, and let I be an (left, right, two-sided) ideal of R. Say that I is small if |I| < |R| and large if |R/I| < |R|. In this paper, we present results on small and large ideals. In particular, we study their interdependence and how they influence the structure of R. Conversely, we investigate how the ideal structure of R determines the existence of small and large ideals.

For a ring R, there are classical facts that R is right Noetherian if and only if every direct sum of injective right R-modules is injective, and R is right Noetherian if and only if every essential extension of a direct sum of injective hulls of simple right R-modules is a direct sum of injective right R-modules. In this paper, we prove that R is right Noetherian if and only if every essential extension of a direct sum of injective hulls of simple right R-modules is a direct sum of either injective right R-modules or projective right R-modules.

Let R ⊆ S be a unital extension of commutative rings, with the integral closure of R in S, such that there exists a finite maximal chain of rings from R to S. Then S is a P-extension of R, is a normal pair, each intermediate ring of R ⊆ S has only finitely many prime ideals that lie over any given prime ideal of R, and there are only finitely many -subalgebras of S. Each chain of rings from R to S is finite if dim(R) = 0; or if R is a Noetherian (integral) domain and S is contained in the quotient field of R; or if R is a one-dimensional domain and S is contained in the quotient field of R; but not necessarily if dim(R) = 2 and S is contained in the quotient field of R. Additional domain-theoretic applications are given.

Many studies have been conducted to characterize commutative rings whose finitely generated modules are direct sums of cyclic modules (called FGC rings), however, the characterization of noncommutative FGC rings is still an open problem, even for duo rings. We study FGC rings in some special cases, it is shown that a local Noetherian ring R is FGC if and only if R is a principal ideal ring if and only if R is a uniserial ring, and if these assertions hold R is a duo ring. We characterize Noetherian duo FGC rings. In fact, it is shown that a duo ring R is a Noetherian left FGC ring if and only if R is a Noetherian right FGC ring, if and only if R is a principal ideal ring.

This paper focusses on the question of when modules have ADS- preenvelopes and covers. For a ring $R$, it is proved that every ADS module over $R$ is injective if and only if every right $R$-module has an ADS-(pre)envelope (an ADS-(pre)cover). In this paper, we also introduce a generalization of ADS modules stated in terms of their invariance under certain automorphisms of their envelopes.

It is well known that a polynomial $f(x)$ over a commutative ring $R$ with identity is a zero-divisor in $R[x]$ if and only if $f(x)$ has a non-zero annihilator in the base ring, where $R[x]$ is the polynomial ring with indeterminate $x$ over $R$. But this result fails in non-commutative rings and in the case of formal power series ring. In this paper, we consider the problem of determining some annihilator properties of the formal power series ring $R[[x]]$ over an associative non-commutative ring $R$. We investigate relations between power series-wise McCoy property and other standard ring-theoretic properties. In this context, we consider right zip rings, right strongly $AB$ rings and rings with right Property $(A)$. We give a generalization (in the case of non-commutative ring) of a classical results related to the annihilator of formal power series rings over the commutative Noetherian rings. We also give a partial answer, in the case of formal power series ring, to the question posed in [1 Question, p. 16].

Let $(R,𝔪)$ be a Noetherian local ring of dimension $d\ge 1$ and let $M$ be a nonzero finitely generated $R$-module. Let $n$ be a positive integer and let $x_1,\dots ,x_n\in 𝔪$ be an $M$-regular sequence. In this paper we shall present some equivalent conditions for the vanishing of the $R$-modules ${Tor}_2^R(R/(x_1,\dots ,x_n),M)$ and ${Ext}_R^{n+2}(R/(x_1,\dots ,x_n),M)$.

Let $R$ be a commutative Noetherian domain, $M$ a nonzero $R$-module of finite injective dimension, and $I$ be a nonzero ideal of $R$. In this paper, we prove that whenever $\mathrm{\text{injdim}}M=cd(I,M)=t$, then the annihilator of $H_I^t(M)$ is zero. Also, we calculate the annihilator of $H_I^{n+t}(N,M)$ for finitely generated $R$-modules $N$ and $M$ with conditions $\mathrm{\text{projdim}}N=n<\infty$ and $\mathrm{\text{injdim}}M=t<\infty$. Moreover, if $(R,𝔪)$ is a regular Noetherian local ring and $𝔭\in \mathrm{\text{Spec}}(R)$ such that $d=\mathrm{\text{dim}}\frac{R}{𝔭}\ge 2$, then we show that there exists an ideal $J$ of $R$ such that $𝔭\subseteq J$, $ht\frac{J}{𝔭}=1$ and $J^n{H}_{𝔪}^1(\frac{R}{{𝔭}^n})=0$.

The purpose of this paper is to study the structure of rings over which every essential extension of a direct sum of a family of simple modules is a direct sum of automorphism-invariant modules. We show that if $R$ is a right quotient finite dimensional (q.f.d.) ring satisfying this property, then $R$ is right Noetherian. Also, we show a von Neumann regular (semiregular) ring $R$ with this property is Noetherian. Moreover, we prove that a commutative ring with this property is an Artinian principal ideal ring.

We study a dimensional-invariant related in some way to the conditions of Krull-Akizuki’s theorem.

Let $(R,𝔪)$ be a one-dimensional commutative Noetherian complete local ring. Assume that the cohomology annihilator $\mathrm{\text{ca}}(R)$ is $𝔪$-primary. We use the notion of Gabriel–Roiter (co)measure in the category of maximal Cohen–Macaulay $R$-modules and prove that, if there is an infinite set ${ℱ}$ of indecomposable maximal Cohen–Macaulay $R$-modules of bounded multiplicity, then there are indecomposable maximal Cohen–Macaulay modules of arbitrary large multiplicity which are cogenerated by modules in ${ℱ}$. This, in particular, guarantees the validity of the first Brauer–Thrall-type theorem for the category of maximal Cohen–Macaulay $R$-modules.