Narrow Results

Let $R$ be a commutative ring with unity. The prime ideal sum graph of the ring $R$ is the simple undirected graph whose vertex set is the set of all nonzero proper ideals of $R$ and two vertices $I$ and $J$ are adjacent if and only if $I+J$ is a prime ideal of $R$. In this paper, we obtain the strong metric dimension of the prime ideal sum graph for various classes of Artinian nonlocal commutative rings.

Let $R$ be a ring with unity. The clean graph $\mathrm{\text{Cl}}(R)$ of a ring $R$ is the simple undirected graph whose vertices are of the form $(e,u)$, where $e$ is an idempotent element and $u$ is a unit of the ring $R$, and two vertices $(e,u)$, $(f,v)$ of $\mathrm{\text{Cl}}(R)$ are adjacent if and only if $ef=fe=0$ or $uv=vu=1$. In this paper, for a commutative ring $R$, first we obtain the strong resolving graph of $\mathrm{\text{Cl}}(R)$ and its independence number. Using them, we determine the strong metric dimension of the clean graph of an arbitrary commutative ring. As an application, we compute the strong metric dimension of $\mathrm{\text{Cl}}(R)$, where $R$ is a commutative Artinian ring.

Let $R$ be a commutative ring with unity. The prime ideal sum graph of the ring $R$ is the simple undirected graph whose vertex set is the set of all nonzero proper ideals of $R$ and two distinct vertices $I$, $J$ are adjacent if and only if $I+J$ is a prime ideal of $R$. In this paper, we characterize all commutative Artinian rings whose prime ideal sum graphs are line graphs. Finally, we give a description of all commutative Artinian rings whose prime ideal sum graph is the complement of a line graph.

For each commutative ring $R$ we associate a simple graph ${\mathrm{\Gamma }}_1^{\ast }(R)$. We investigate the properties of an Artinian ring $R$ when ${\mathrm{\Gamma }}_1^{\ast }(R)$ is a star.

Given a commutative ring with identity $A$ and an $A$-module $M$, a subset $H$ of $M$ is a cyclic covering of $M$, if this module is the union of the cyclic submodules $[h]=\{ah:a\in A\}$, where $h\in H$. Such covering is said to be irredundant, if no proper subset of $H$ is a cyclic covering of $M$. In this work, an irredundant cyclic covering of $A^n$ is constructed for every Artinian commutative ring $A$. As a consequence, a cyclic covering of minimal cardinality of $A^n$ is obtained for every finite commutative ring $A$, extending previous results in the literature.

For a space, we investigate its CJL (cohomology jump loci), sitting inside varieties of representations of the fundamental group. To do this, for a CDG (commutative differential graded) algebra, we define its CJL, sitting inside varieties of flat connections. The analytic germs at the origin 1 of representation varieties are shown to be determined by the Sullivan 1-minimal model of the space. Up to a degree q, the two types of CJL have the same analytic germs at the origins, when the space and the algebra have the same q-minimal model. We apply this general approach to formal spaces (obtaining the degeneration of the Farber–Novikov spectral sequence), quasi-projective manifolds, and finitely generated nilpotent groups. When the CDG algebra has positive weights, we elucidate some of the structure of (rank one complex) topological and algebraic CJL: all their irreducible components passing through the origin are connected affine subtori, respectively rational linear subspaces. Furthermore, the global exponential map sends all algebraic CJL into their topological counterpart.

Given a ring R, we study the bimodules M for which the trivial extension R ∝ M is morphic. We obtain a complete characterization in the case where R is left perfect, and we prove that R ∝ Q/R is morphic when R is a commutative reduced ring with classical ring of quotients Q. We also extend some known results concerning the connection between morphic rings and unit regular rings.

Let R be a ring, R is called a right ACS ring if the right annihilator of each element is essential in a direct summand of R. We prove that right CF ring is right artinian under right ACS condition. In particular, FGF conjecture is true under right ACS condition. We also give some conditions under which right noetherian left P-injective and left ACS ring is QF.

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.

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 $J^n{H}_{𝔪}^1(\frac{R}{{𝔭}^n})=0$.

Let $R$ be a commutative ring with nonzero identity and $S\subseteq R$ be a multiplicatively closed subset. In this paper, we study $S$-Artinian rings and finitely $S$-cogenerated rings. A commutative ring $R$ is said to be an $S$-Artinian ring if for each descending chain of ideals ${\{I_n\}}_{n\in \mathbb{N}}$ of $R,$ there exist $s\in S$ and $k\in \mathbb{N}$ such that $s{I}_k\subseteq I_n$ for all $n\ge k.$ Also, $R$ is called a finitely $S$-cogenerated ring if for each family of ideals ${\{I_{\alpha }\}}_{\alpha \in \mathrm{\Delta }}$ of $R,$ where $\mathrm{\Delta }$ is an index set, ${\bigcap }_{\alpha \in \mathrm{\Delta }}{I}_{\alpha }=0$ implies $s({\bigcap }_{\alpha \in {\mathrm{\Delta }}^{\prime }}{I}_{\alpha })=0$ for some $s\in S$ and a finite subset ${\mathrm{\Delta }}^{\prime }\subseteq \mathrm{\Delta }.$ Moreover, we characterize some special rings such as Artinian rings and finitely cogenerated rings. Also, we extend many properties of Artinian rings and finitely cogenerated rings to $S$-Artinian rings and finitely $S$-cogenerated rings.

It is proved that a ring A is a right or left Noetherian, right distributive, centrally essential ring if and only if $A=A_1\times \cdots \times A_n$, where each of the rings $A_i$ is either a commutative Dedekind domain or a left and right Artinian, left and right uniserial ring.

Recall that a ring $R$ is a Hilbert ring if any maximal ideal of $R[X]$ contracts to a maximal ideal of $R$. The main purpose of this paper is to characterize the prime ideals of a commutative ring $R$ which are traces of the maximal ideals of the polynomial ring $R[X]$. In this context, we prove that if $p$ is a prime ideal of $R$ such that $R/p$ is a semi-local domain of (Krull) dimension $\le 1$, then $p$ is the trace of a maximal ideal of $R[X]$. Whereas, if $R$ is Noetherian and either ($dim(R/p)\ge 2$) or (the quotient field of $R/p$ is algebraically closed, $dim(R/p)=1$ and $R/p$ is not semi-local), then $p$ is never the trace of a maximal ideal of $R[X]$. Putting these results into use in investigating the Ext-index of Noetherian rings, we establish connections between the finiteness of the Ext-index of localizations of the polynomial rings $R[X]$ and the finiteness of the Ext-index of localizations of $R$. This allows us to provide a new class of rings satisfying some known conjectures on Ext-index of Noetherian rings as well as to build bridges between these conjectures.

A proper subring S of a ring R is said to be maximal if there is no subring of R properly between S and R. If R is a noetherian domain with |R| > 2^ℵ₀, then |Max(R)| ≤ |RgMax(R)|, where RgMax(R) is the set of maximal subrings of R. A useful criterion for the existence of maximal subrings in any ring R is also given. It is observed that if S is a maximal subring of a ring R, then S is artinian if and only if R is artinian and integral over S. Surprisingly, it is shown that any infinite direct product of rings has always maximal subrings. Finally, maximal subrings of zero-dimensional rings are also investigated.

Let R be a commutative ring and 𝔸(R) be the set of ideals with non-zero annihilators. The annihilating-ideal graph of R is defined as the graph 𝔸𝔾(R) with the vertex set 𝔸(R)* = 𝔸(R)\{(0)} and two distinct vertices I and J are adjacent if and only if IJ = (0). Here, we present some results on the clique number and the chromatic number of the annihilating-ideal graph of a commutative ring. It is shown that if R is an Artinian ring and ω (𝔸𝔾(R)) = 2, then R is Gorenstein. Also, we investigate commutative rings whose annihilating-ideal graphs are complete or bipartite.

Let R be a commutative ring. In this paper, we develop the existence of direct limits of finite products of fields as subrings of R.

In [8] M. Harada studied a left artinian ring R such that every non-small left R-module contains a non-zero injective submodule. (We can see the results also in his lecture note [9, §10.2].) In [10] K. Oshiro called the ring a left H-ring and later in [11] he called it a left Harada ring. Since then many significant results are invented. We can see many results on left Harada rings in [4] and many equivalent conditions in [3, Theorem B]. In [5] we introduce “H-epimorphism” and “co-H-sequence” and, in a two-sided Harada ring, we characterize the structure of a right Harada ring using a well-indexed set of a left Harada ring. Further in [6] we introduce another new concept “weak co-H-sequence” and study two-sided Harada ring. In this paper, from a given QF ring, we construct two-sided Harada rings.