Ring-theoretic Properties Research Articles

Ring–theoretic properties of locally Mori domains and rings of the form $$\mathrm {Int}(E,D)$$

Ring–theoretic properties of locally Mori domains and rings of the form $$\mathrm {Int}(E,D)$$

Ring theoretical properties of epsilon-strongly graded rings and Leavitt path algebras

We extend Dade’s theorem to the realm of nearly epsilon-strongly graded rings, and present new characterizations of strongly and epsilon-strongly graded rings. Further, we give conditions for an epsilon-strongly graded ring to be written as a direct sum of strongly graded rings and a ring with trivial gradation. Our results are applied to characterize strongly graded Leavitt path algebras endowed with the canonical gradation. Finally we show that for any there is a Leavitt path algebra which is a sum of n − 1 strongly graded rings and a ring with trivial gradation.

On a spanning subgraph of the annihilating-ideal graph of a commutative ring

The rings considered in this paper are commutative with identity which are not integral domains. Let [Formula: see text] be a ring. Let us denote the set of all annihilating ideals of [Formula: see text] by [Formula: see text] and [Formula: see text] by [Formula: see text]. With [Formula: see text], we associate an undirected graph, denoted by [Formula: see text], whose vertex set is [Formula: see text] and distinct vertices [Formula: see text] and [Formula: see text] are adjacent in this graph if and only if [Formula: see text] and [Formula: see text]. The aim of this paper is to study the interplay between the graph-theoretic properties of [Formula: see text] and the ring-theoretic properties of [Formula: see text].

Ideal-based quasi zero divisor graph

Let $R$ be a commutative ring with identity and $I$ a proper ideal of $R$. In this paper we introduce the ideal-based quasi zero divisor graph $Q\Gamma_{I}(R)$ of $R$ with respect to $I$ which is an undirected graph with vertex set $V=\{a\in R\backslash\sqrt{I}:$ $ab\in I$ for some $b\in R\backslash\sqrt{I}\}$ and two distinct vertices $a$ and $b$ are adjacent if and only if $ab\in I$. We study the basic properties of this graph such as diameter, girth, dominaton number, etc. We also investigate the interplay between the ring theoretic properties of a Noetherian multiplication ring $R$ and the graph-theoretic properties of $Q\Gamma_{I}(R)$.

Finite étale extensions of Tate rings and decompletion of perfectoid algebras

In this paper, we examine the behavior of ideal-adic separatedness and completeness under certain ring extensions using trace map. Then we prove that adic completeness of a base ring is hereditary to its ring extension under reasonable conditions. We aim to give many results on ascent and descent of certain ring theoretic properties under completion. As an application, we give conceptual details to the proof of the almost purity theorem for Witt-perfect rings by Davis and Kedlaya. Witt-perfect rings have the advantage that one does not need to assume that the rings are complete and separated.

English

This paper aims at the study of the notions of periodic, UU and semiclean properties in various context of commutative rings such as trivial ring extensions, amalgamations and pullbacks. The results obtained provide new original classes of rings subject to various ring theoretic properties.

On generalized zero-divisor graphs of a non-commutative ring with respect to an ideal

Let R be a non-commutative ring, and I be an ideal of R. In this paper, we generalize the definition of the zero-divisor graph of R with respect to I, and define several generalized zero-divisor graphs of R with respect to I. In this paper, we investigate the ring-theoretic properties of R and the graph-theoretic properties of all the generalized zero-divisor graphs. We study some basic properties of these generalized zero-divisor graphs related to the connectedness, the diameter and the girth. We also investigate some properties of these generalized zero-divisor graphs with respect to primal ideals.

Integral domains issued from associated primes

For a property of integral domains, D is said to be an - domain if DP has the property for all associated prime ideals P of an integral domain D. Our first goal is to study some ring-theoretic properties of - domains; in particular, we investigate the property of being an - domain in the polynomial ring and the t-flat overring extension. Also we study integral domains satisfying the ascending chain condition on associated prime ideals.

Structure and isomorphisms of quantum generalized Heisenberg algebras

In [S. A. Lopes and F. Razavinia, Quantum generalized Heisenberg algebras and their representations, preprint (2020), arXiv:2004.09301] we introduced a new class of algebras, which we named quantum generalized Heisenberg algebras and which depend on a parameter [Formula: see text] and two polynomials [Formula: see text]. We have shown that this class includes all generalized Heisenberg algebras (as defined in [E. M. F. Curado and M. A. Rego-Monteiro, Multi-parametric deformed Heisenberg algebras: A route to complexity, J. Phys. A: Math. Gen. 34(15) (2001) 3253; R. Lü and K. Zhao, Finite-dimensional simple modules over generalized Heisenberg algebras, Linear Algebra Appl. 475 (2015) 276–291, MR 3325233]) as well as generalized down-up algebras (as defined in [G. Benkart and T. Roby, Down-up algebras, J. Algebra 209(1) (1998) 305–344; T. Cassidy and B. Shelton, Basic properties of generalized down-up algebras, J. Algebra 279(1) (2004) 402–421, MR 2078408 (2005f:16051)]), but the parameters of freedom we allow for give rise to many algebras which are in neither one of these two classes. Having classified their finite-dimensional irreducible representations in [S. A. Lopes and F. Razavinia, Quantum generalized Heisenberg algebras and their representations, preprint (2020), arXiv:2004.09301], in this paper, we turn to their classification by isomorphism, the description of their automorphism groups and the study of their ring-theoretical properties.

A survey on the Intersection graphs of ideals of rings

Let L(R) denote the set of all non-trivial left ideals of a ring R. The intersection graph of ideals of a ring R is an undirected simple graph denoted by G(R) whose vertices are in a one-to-one correspondence with L(R) and two distinct vertices are joined by an edge if and only if the corresponding left ideals of R have a non-zero intersection. The ideal structure of a ring reflects many ring theoretical properties. Thus much research has been conducted last few years to explore the properties of G(R). This is a survey of the developments in the study on the intersection graphs of ideals of rings since its introduction in 2009.

Some results on t–locally Strong Mori domains and their rings of integer–valued polynomials

An integral domain D is said to be t–locally Strong Mori (for short, t–LSM) if $$D_\mathfrak {m}$$ is Strong Mori for all t–maximal ideals $$\mathfrak {m}$$ of D. This paper studies some ring–theoretic properties of t–LSM domains and the algebra structure of rings of integer–valued polynomials arising from t–LSM domains. Among other things, we investigate the property of being a t–LSM domain in the t–flat overring extension, the t–Nagata ring, the polynomial ring, pullback construction and the power series ring. Also, we study $${\mathrm {Int}}(D)$$ over a t–LSM domain D. Precisely, we are interested in the Krull dimension, the trivial case and some module structure properties.

Descent of properties of rings and pairs of rings to fixed rings

Let G be a group acting via ring automorphisms on an integral domain R. A ring-theoretic property of R is said to be G-invariant, if \(R^G\) also has the property, where \(R^G=\{r\in R \ | \ \sigma (r)=r \ \text {for all} \ \sigma \in G\},\) the fixed ring of the action. In this paper we prove the following classes of rings are invariant under the operation \(R\rightarrow R^G:\) locally pqr domains, Strong G-domains, G-domains, Hilbert rings, S-strong rings and root-closed domains. Further let \(\mathscr {P}\) be a ring theoretic property and \(R\subseteq S\) be a ring extension. A pair of rings (R, S) is said to be a \(\mathscr {P}\)-pair, if T satisfies \(\mathscr {P}\) for each intermediate ring \(R\subseteq T\subseteq S.\) We also prove that the property \(\mathscr {P}\) descends from \((R,S)\rightarrow (R^G, S^G)\) in several cases. For instance, if \(\mathscr {P}=\) Going-down, Pseudo-valuation domain and “finite length of intermediate chains of domains”, we show each of these properties successfully transfer from \((R,S)\rightarrow (R^G, S^G).\)

A comparison of group algebras of dihedral and quaternion groups

The group algebras of the generalised quaternion groups and the dihedral groups of order a power of 2 are compared. Their group algebras over a finite field of characteristic 2 are known to be non-isomorphic and several new proofs of this are given which may be of independent interest. However, the two group algebras are very similar and are shown to have many ring theoretic properties in common. Lastly, the semisimple case (where the characteristic of the field is greater than 2) is considered and the minimum noncommutative counterexample to the Isomorphism Problem is identified.

Twisted Polynomial and Power Series Rings

Let R be a commutative ring with identity and $${\mathbb {N}}_0$$ be the additive monoid of nonnegative integers. We say that a function $$t : {\mathbb {N}}_0 \times {\mathbb {N}}_0 \rightarrow R$$ is a twist function on R if t satisfies the following three properties for all $$n, m, q \in {\mathbb {N}}_0$$ : (i) $$t(0,q) = 1$$ , (ii) $$t(n,m) = t(m,n)$$ , and (iii) $$t(n,m) \cdot t(n + m, q) = t (n, m + q) \cdot t(m, q)$$ . Let $$R[\![X]\!]$$ (resp., R[X]) be the set of power series (resp., polynomials) with coefficients in R. For $$f = \sum _{n=0}^{\infty } a_nX^n$$ and $$g = \sum _{n=0}^{\infty } b_nX^n \in R[\![X]\!]$$ , let $$f+g = \sum _{n=0}^{\infty } (a_n+b_n)X^n$$ , $$f*_tg = \sum _{n=0}^{\infty }(\sum _{i+j = n}t(i,j)a_ib_j)X^n$$ . Then, $$R^t[\![X]\!]:= (R[\![X]\!], +, *_t)$$ and $$R^t[X] := (R[X], +, *_t)$$ are commutative rings with identity that contain R as a subring. In this paper, we study ring-theoretic properties of $$R^t[\![X]\!]$$ and $$R^t[X]$$ with focus on divisibility properties including UFDs and GCD-domains. We also show how these two rings are related to the usual power series and polynomial rings.

Reversibility of skew Hurwitz series rings

We study the reversibility of skew Hurwitz series at zero as a generalization of an $\alpha$-rigid ring, introducing the concept of skew Hurwitz reversibility. A ring $R$ is called skew Hurwitz reversible ($SH$-reversible, for short), if the skew Hurwitz series ring $(HR,\alpha)$ is reversible i.e. whenever skew Hurwitz series $f, g\in (HR,\alpha)$ satisfy $fg=0$, then $gf=0$. We examine some characterizations and extensions of $SH$-reversible rings in relation with several ring theoretic properties which have roles in ring theory.

Eigenvalues of zero-divisor graphs of finite commutative rings

We investigate eigenvalues of the zero-divisor graph Gamma (R) of finite commutative rings R and study the interplay between these eigenvalues, the ring-theoretic properties of R and the graph-theoretic properties of Gamma (R). The graph Gamma (R) is defined as the graph with vertex set consisting of all nonzero zero-divisors of R and adjacent vertices x, y whenever xy = 0. We provide formulas for the nullity of Gamma (R), i.e., the multiplicity of the eigenvalue 0 of Gamma (R). Moreover, we precisely determine the spectra of Gamma ({mathbb {Z}}_p times {mathbb {Z}}_p times {mathbb {Z}}_p) and Gamma ({mathbb {Z}}_p times {mathbb {Z}}_p times {mathbb {Z}}_p times {mathbb {Z}}_p) for a prime number p. We introduce a graph product times _{Gamma } with the property that Gamma (R) cong Gamma (R_1) times _{Gamma } cdots times _{Gamma } Gamma (R_r) whenever R cong R_1 times cdots times R_r. With this product, we find relations between the number of vertices of the zero-divisor graph Gamma (R), the compressed zero-divisor graph, the structure of the ring R and the eigenvalues of Gamma (R).

A note on $\lambda$-domains and $\Delta$-domains

Let $R$ be an integral domain. Then $R$ is said to be a $\lambda$-domain if the set of all overrings of $R$ is linearly ordered by inclusion. If $R_1 + R_2$ is an overring of $R$ for each pair of overrings $R_1, R_2$ of $R$, then $R$ is said to be a $\Delta$-domain. We show that if $R\subset T$ is an extension of integral domains such that each proper subring of $T$ containing $R$ is a $\lambda$-domain (resp., $\Delta$-domain), then $T$ is a $\lambda$-domain (resp., $\Delta$-domain under some conditions). Moreover, the pair $(R, T)$ is a residually algebraic pair. Two new ring theoretic properties, namely $\lambda$-property of domains and $\Delta$-property of domains are introduced and studied.

Computational aspects of Burnside rings part II: important maps

The Burnside ring $$\mathcal {B}(G)$$ of a finite group G, a classical tool in the representation theory of finite groups, is studied from the point of view of computational algebra. In the first part (cf. Kreuzer and Patil, Beitr Algebra Geom 58:427–452, 2017) we examined the ring theoretic properties of $$\mathcal {B}(G)$$ using the methods of computer algebra. In this part we shift our focus to important maps between two Burnside rings and make several well-known maps of representation theory explicitly computable. The inputs of all our algorithms are the tables of marks of the two groups, and the outputs are matrices of integers representing the maps via their image in the ghost ring. All algorithms have been implemented in the computer algebra system ApCoCoA and are illustrated by applying them to explicit examples. Especially, we study the restriction and induction maps, the projection and inflation maps, the conjugation isomorphism, and the Frobenius-Wielandt homomorphism.

Ring-theoretic (In)finiteness in reduced products of Banach algebras

AbstractWe study ring-theoretic (in)finiteness properties—such as Dedekind-finiteness and proper infiniteness—of ultraproducts (and more generally, reduced products) of Banach algebras.While we characterise when an ultraproduct has these ring-theoretic properties in terms of its underlying sequence of algebras, we find that, contrary to the $C^*$ -algebraic setting, it is not true in general that an ultraproduct has a ring-theoretic finiteness property if and only if “ultrafilter many” of the underlying sequence of algebras have the same property. This might appear to violate the continuous model theoretic counterpart of Łoś’s Theorem; the reason it does not is that for a general Banach algebra, the ring theoretic properties we consider cannot be verified by considering a bounded subset of the algebra of fixed bound. For Banach algebras, we construct counter-examples to show, for example, that each component Banach algebra can fail to be Dedekind-finite while the ultraproduct is Dedekind-finite, and we explain why such a counter-example is not possible for $C^*$ -algebras. Finally, the related notion of having stable rank one is also studied for ultraproducts.

Some Remarks on the Annihilator Graph of a Commutative Ring

Let $R$ be a commutative ring with nonzero identity. The annihilator graph of $R$, denoted by $AG(R)$, is the (undirected) graph whose vertex set is the set of all nonzero zero-divisors of $R$ and two distinct vertices $x$ and $y$ are adjacent if and only if $\ann_R(xy)\neq {\rm ann}_R(x)\cup {\rm ann}_R(y)$. We investigate the interplay between ring-theoretic properties of $R$ and graph-theoretic properties of $AG(R)$. We study the relation between two graphs $\Gamma(R)$ and $AG(R)$, where $R$ is a non-reduced commutative ring. Also, we completely characterize the rings whose annihilator graphs are complete.