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.

In this paper, we study endo-reduced modules as modules whose endomorphism rings have no nonzero nilpotent elements. We characterize their properties for different classes of modules, including ${𝒦}$-non-singular modules, multiplication modules and finitely generated modules over commutative Dedekind domains. In the subcategory of finitely generated modules, it is shown that the class of rings $R$ for which every faithful multiplication $R$-module is endo-reduced is precisely that of reduced rings; while the class of rings $R$ for which every multiplication $R$-module is endo-reduced is precisely that of von Neumann regular rings. Characterizations of when an endo-reduced module will be a reduced module are given. We prove that a finitely generated module over a principal ideal domain (PID) is endo-reduced exactly if it is either a semisimple module with pair-wise non-isomorphic submodules or a torsion-free module which is isomorphic to the underlying ring.

Let $R$ be a ∗-ring. Then $R$ is called a ∗-symmetric ring if for any $a,b,c\in R,abc=0$ implies $ac{b}^{\ast }=0.$ Obviously, ∗-symmetric ring is certainly symmetric ring, while ∗-ring and symmetric ring does not equal ∗-symmetric ring. Even though reduced rings are symmetric, reduced rings need not be ∗-symmetric ring. In this paper, we further add some conditions for symmetric ring to be ∗-symmetric and reduced. As an application, we discover if $R$ be a ∗-symmetric ring, then $R^{\#}=R^{EP}=R^{+}=R^{\mathrm{\text{reg}}}.$ Also, we expand the definition of ∗-symmetric ring to different combinations of $a,b,c$. The properties of ∗-symmetric ring play a role in generalized inverses of elements in 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.

Jacobson introduced the concept of K-rings, continuing the investigation of Kaplansky and Herstein into the commutativity of rings. In this note we focus on the ring-theoretic properties of K-rings. We first construct basic examples of K-rings to be handled easily. It is shown that a semiprime K-ring of bounded index of nilpotency is a commutative domain. It is proved that if R is a prime K-ring then its classical quotient ring is a local ring with a nil Jacobson radical. We also show that if R is a π-regular K-ring then R/P is a field for every strongly prime ideal P of R. The basic structure of a condition, unifying K-rings and reversible rings, is studied with respect to zero-divisors in matrices and polynomials.

Let R be a ring and G a group of automorphisms of R. In this paper we investigate the transfer of the Von Neumman regularity from the total quotient ring of R to the total quotient ring of its fixed ring. We prove also that, for a locally finite group, Min(R^G) inherits the compactness from Min(R), where Min(R) denotes the set of all minimal prime ideals of R. Finally, We explore some conditions under which we can transfer the PF-property (respectively, pseudo PF-property) from a ring to its fixed subring.

Results of Davis on normal pairs (R, T) of domains are generalized to (commutative) rings with nontrivial zero-divisors, particularly complemented rings. For instance, if T is a ring extension of an almost quasilocal complemented ring R, then (R, T) is a normal pair if and only if there is a prime ideal P of R such that T = R_[P], R/P is a valuation domain and PT = P. Examples include sufficient conditions for the "normal pair" property to be stable under formation of infinite products and ⋈ constructions.

Let R be a reduced commutative ring with 1 ≠ 0. Then R is a partially ordered set under the Abian order defined by x ≤ y if and only if xy = x². Let R_E be the set of equivalence classes for the equivalence relation on R given by x ~ y if and only if ann_R(x) = ann_R(y). Then R_E is a commutative Boolean monoid with multiplication [x][y] = [xy] and is thus partially ordered by [x] ≤ [y] if and only if [xy] = [x]. In this paper, we study R and R_E as both monoids and partially ordered sets. We are particularly interested in when R_E can be embedded in R as either a monoid or a partially ordered set.

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).

If $A\subseteq B$ are (commutative) rings, $[A,B]$ denotes the set of intermediate rings and $(A,B)$ is called an almost valuation (AV)-ring pair if each element of $[A,B]$ is an AV-ring. Many results on AV-domains and their pairs are generalized to the ring-theoretic setting. Let $R\subseteq S$ be rings, with ${\overline{R}}_S$ denoting the integral closure of $R$ in $S$. Then $(R,S)$ is an AV-ring pair if and only if both $(R,{\overline{R}}_S)$ and $({\overline{R}}_S,S)$ are AV-ring pairs. Characterizations are given for the AV-ring pairs arising from integrally closed (respectively, integral; respectively, minimal) ring extensions $R\subseteq S$. If $(R,S)$ is an AV-ring pair, then $R\subseteq S$ is a P-extension. The AV-ring pairs $(R,S)$ arising from root extensions are studied extensively. Transfer results for the “AV-ring” property are obtained for pullbacks of $(B,I,D)$ type, with applications to pseudo-valuation domains, integral minimal ring extensions, and integrally closed maximal non-AV subrings. Several sufficient conditions are given for $(R,S)$ being an AV-ring pair to entail that $S$ is an overring of $R$, but there exist domain-theoretic counter-examples to such a conclusion in general. If $(R,S)$ is an AV-ring pair and $R\subseteq S$ satisfies FCP, then each intermediate ring either contains or is contained in ${\overline{R}}_S$. While all AV-rings are quasi-local going-down rings, examples in positive characteristic show that an AV-domain need not be a divided domain or a universally going-down domain.

We obtain a common generalization of the results by Wong and Birkenmeier-Kim-Park, respectively, which say that a reduced ring with unity is strongly (respectively, weakly) regular if and only if all of its prime homomorphic images are division rings (respectively, simple domains). Our arguments are different from those in the known proofs and are quite simple. They also give a characterization of weakly regular reduced rings without unity. This characterization implies in particular that the class of weakly regular reduced rings forms a radical class. However, even if a weakly regular reduced ring has no unity, its prime homomorphic images must be simple domains with unity. In the second part of the paper, we study reduced rings whose prime homomorphic images are simple domains (not necessarily with unity).

Let $R_n$ be a finite reduced ring with $n$ maximal ideals ${𝔪}_i$ and let $\mathrm{\Gamma }(R_n)$ be the zero-divisor graph associated to $R_n.$ The class of rings $R_n$ contains the Boolean rings as a subclass. When $R_n/{𝔪}_i=𝔽$ for all $i$ where $𝔽$ is a finite field, we associate two $(n-1)\times (n-1)$ sized matrices $P$ and $Q$ to the graph $\mathrm{\Gamma }(R_n)$ having combinatorial entries and use these matrices to determine the spectrum of this graph. More precisely, we show that every eigenvalue of $P$ and of $-Q$ is an eigenvalue of $\mathrm{\Gamma }(R_n).$ To do this, we give a recursive description of the adjacency matrix of this graph and also exhibit its equitable partition. This is used in computing the determinant, rank and nullity of the adjacency matrix. Further, we propose that the eigenvalues of $P,$$-Q$ and the eigenvalue $0$ exhaust all the eigenvalues of $\mathrm{\Gamma }(R_n).$

In this paper, we introduce power-serieswise McCoy rings, which are a generalization of power-serieswise Armendariz rings, and investigate their properties. We show that a ring R is power-serieswise McCoy if and only if the ring consisting of n × n upper triangular matrices with equal diagonal entries over R is power-serieswise McCoy. We also prove that a direct product of rings is power-serieswise McCoy if and only if each of its factors is power-serieswise McCoy. Meanwhile we show that power-serieswise McCoy rings may be neither semi-commutative nor power-serieswise Armendariz.

It is well known that the m × m upper triangular matrix ring over any ring is not ZI_n (and so not ZC_n) for m ≥ 2. In this paper, we find some ZC_n subrings and ZI_n subrings of the upper triangular matrix ring over a reduced ring.

It is shown that if R is a ring with unit element which is not algebraic over the prime subring of R, then R has a maximal subring. It is shown that whenever R ⊆ T are rings such that there exists a maximal subring V of T, which is integrally closed in T and U(R) ⊈ V, then R has a maximal subring. In particular, it is proved that if R is algebraic over ℤ and there exists a natural number n > 1 with n ∈ U(R), then R has a maximal subring. It is shown that if R is an infinite direct product of certain fields, then the maximal ideals M for which R_M (R/M) has maximal subrings are characterized. It is observed that if R is a ring, then either R has a maximal subring or it must be a Hilbert ring. In particular, every reduced ring R with |R|>2^2ℵ₀ or J(R) ≠ 0 has a maximal subring. Finally, the semi-local rings having maximal subrings are fully characterized.

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).

It is proved that for matrices $A$, $B$ in the $n$ by $n$ upper triangular matrix ring $T_n(R)$ over a domain $R$, if $AB$ is nonzero and central in $T_n(R)$ then $AB=BA$. The $n$ by $n$ full matrix rings over right Noetherian domains are also shown to have this property. In this article we treat a ring property that is a generalization of this result, and a ring with such a property is said to be weakly reversible-over-center. The class of weakly reversible-over-center rings contains both full matrix rings over right Noetherian domains and upper triangular matrix rings over domains. The structure of various sorts of weakly reversible-over-center rings is studied in relation to the questions raised in the process naturally. We also consider the connection between the property of being weakly reversible-over-center and the related ring properties.

For a ring endomorphism α, we introduce and investigate skew Hurwitz serieswise Armendariz (or SHA) rings which are a generalization of α-rigid rings and determine the radicals of the skew Hurwitz series ring (HR, α), in terms of those of R. We prove that several properties transfer between R and the extensions, in case R is an SHA-ring. We also construct various types of nonreduced SHA-rings.

Let $\overline{\mathrm{\Gamma }(R)}$ be the complement of the zero-divisor graph of a finite commutative ring $R$. In this paper, we provide the answer of the question (ii) raised by Osba and Alkam in [11] and prove that $\overline{\mathrm{\Gamma }(R)}$ is a divisor graph if $R$ is a local ring. It is shown that when $R$ is a product of two local rings, then $\overline{\mathrm{\Gamma }(R)}$ is a divisor graph if one of them is an integral domain. Further, if $|Ass(R)|=2$, then $\overline{\mathrm{\Gamma }(R)}$ is a divisor graph.

For a finite undirected looped graph $\mathring{G}$, the universal adjacency matrix $U(\mathring{G})$ is a linear combination of the adjacency matrix $A(\mathring{G})$, the degree matrix $D(\mathring{G})$, the identity matrix $I$ and the all-ones matrix $J$, that is $U(\mathring{G})=\alpha A(\mathring{G})+\beta D(\mathring{G})+\gamma I+\eta J$, where $\alpha ,\beta ,\gamma ,\eta \in ℝ$ and $\alpha \ne 0$. For a finite commutative ring $R$ with unity, the looped zero divisor graph $\mathring{\mathrm{\Gamma }}(R)$ is an undirected graph with the set of all nonzero zero divisors of $R$ as vertices and two vertices (not necessarily distinct) $x$ and $y$ are adjacent if and only if $xy=0$. In this paper, we study some structural properties of $\mathring{\mathrm{\Gamma }}(R)$ by defining an equivalence relation on its vertex set. Then we obtain the universal adjacency eigenpairs of $\mathring{\mathrm{\Gamma }}(R)$, and as a consequence several spectra like the adjacency, Seidel, Laplacian, signless Laplacian, normalized Laplacian, generalized adjacency and convex linear combination of the adjacency and degree matrix of $\mathring{\mathrm{\Gamma }}(R)$ can be obtained in a unified way. Moreover, we get the structural properties and the universal adjacency eigenpairs of the looped zero divisor graph of a reduced ring in a simpler form.