Commutative Ring Research Articles

Minimal Prime and Semiprime Submodules

Prime and semiprime submodules are important generalizations of prime and semiprime ideals to module theory over commutative rings. However, minimal or smallest prime/semiprime submodules have received comparatively less attention. This paper furthers the theory of minimal prime and minimal semiprime submodules through several new directions. After formally introducing these concepts, we establish characterization theorems for minimal prime submodules based on intersections of primes and associated primes. A complete structure theory is also developed for minimal semiprime submodules, allowing their description up to isomorphism. We introduce prime and semiprime operators which stably preserve primeness properties between modules. Results on the persistence of minimal primeness of submodules under such operators are proved. Connections of our minimal semiprime characterization back to classic ring theory are highlighted, recovering past results on minimal prime ideals. Overall, this paper significantly expands the foundations of minimal prime and semiprime submodules, while opening up many new questions at this nexus of module and ring theory.

Signless Laplacian spectrum of the cozero-divisor graph of the commutative ring ℤ𝑛

Abstract Let 𝑅 be a commutative ring with identity 1 ≠ 0 1\neq 0 and let Z ⁢ ( R ) ′ Z(R)^{\prime} be the set of all non-zero and non-unit elements of ring 𝑅. Further, Γ ′ ⁢ ( R ) \Gamma^{\prime}(R) denotes the cozero-divisor graph of 𝑅, is an undirected graph with vertex set Z ⁢ ( R ) ′ Z(R)^{\prime} , and w ∉ z ⁢ R w\notin zR and z ∉ w ⁢ R z\notin wR if and only if two distinct vertices 𝑤 and 𝑧 are adjacent, where q ⁢ R qR is the ideal generated by the element 𝑞 in 𝑅. In this paper, we find the signless Laplacian eigenvalues of the graphs Γ ′ ⁢ ( Z n ) \Gamma^{\prime}(\mathbb{Z}_{n}) for n = p 1 N ⁢ p 2 ⁢ p 3 n=p_{1}^{N}p_{2}p_{3} and p 1 N ⁢ p 2 M ⁢ p 3 p_{1}^{N}p_{2}^{M}p_{3} , where p 1 , p 2 , p 3 p_{1},p_{2},p_{3} are distinct primes and N , M N,M are positive integers. We also show that the cozero-divisor graph Γ ′ ⁢ ( Z p 1 ⁢ p 2 ) \Gamma^{\prime}(\mathbb{Z}_{p_{1}p_{2}}) is a signless Laplacian integral.

Comparable overrings of a commutative ring

Comparable overrings of a commutative ring

Vector-circulant matrices and codes over rings

Circulant matrices are of interest on their own as well as for their relevance to several areas of mathematics. In this note vector-circulant matrices over commutative rings are introduced and related results are given. Vector-circulant based additive codes are presented over a particular finite ring.

On k-semi-centralizing maps of generalized matrix algebras

Abstract Let 𝒢 = 𝒢 (A, M, N, B) be a generalized matrix algebra over a commutative ring with unity. In the present article, we study k-semi-centralizing maps of generalized matrix algebras.

Quasi-Primary Spectrum of a Commutative Ring and a Sheaf of Rings

In this work, the set of quasi-primary ideals of a commutative ring with identity is equipped with a topology and is called the quasi-primary spectrum. Some topological properties of this space are examined. Further, a sheaf of rings on the quasi-primary spectrum is constructed and it is shown that this sheaf is the direct image sheaf with respect to the inclusion map from the prime spectrum of a ring to the quasi-primary spectrum of the same ring.

Dual Zariski Spaces of Modules

Let [Formula: see text] be a commutative ring with identity, [Formula: see text] be an [Formula: see text]-module, [Formula: see text] denote the set of all submodules of [Formula: see text] and [Formula: see text]. For any submodule [Formula: see text] of [Formula: see text], we set [Formula: see text] and [Formula: see text]. Consider [Formula: see text], where [Formula: see text] is the set of all ideals of [Formula: see text]. We set [Formula: see text] and [Formula: see text] for any ideal [Formula: see text] of [Formula: see text]. In this paper, we investigate when, for arbitrary [Formula: see text] and [Formula: see text] as above, [Formula: see text] and [Formula: see text] form a topology and a semimodule, respectively. We investigate the structure of [Formula: see text] in the case that it is a semimodule.

Realizing orders as group rings

An order is a commutative ring that as an abelian group is finitely generated and free. A commutative ring is reduced if it has no non-zero nilpotent elements. In this paper we use a new tool, namely, the fact that every reduced order has a universal grading, to answer questions about realizing orders as group rings. In particular, we address the Isomorphism Problem for group rings in the case where the ring is a reduced order. We prove that any non-zero reduced order R can be written as a group ring in a unique “maximal” way, up to isomorphism. More precisely, there exist a ring A and a finite abelian group G, both uniquely determined up to isomorphism, such that R≅A[G] as rings, and such that if B is a ring and H is a group, then R≅B[H] as rings if and only if there is a finite abelian group J such that B≅A[J] as rings and J×H≅G as groups. Computing A and G for given R can be done by means of an algorithm that is not quite polynomial-time. We also give a description of the automorphism group of R in terms of A and G.

Some notes on the algebraic structure of linear recurrent sequences

AbstractSeveral operations can be defined on the set of all linear recurrent sequences, such as the binomial convolution (Hurwitz product) or the multinomial convolution (Newton product). Using elementary techniques, we prove that this set equipped with the termwise sum and the aforementioned products is an R-algebra, given any commutative ring R with identity. Moreover, we provide explicitly a characteristic polynomial of the Hurwitz product and Newton product of any two linear recurrent sequences. Finally, we also investigate whether these R-algebras are isomorphic, considering also the R-algebras obtained using the Hadamard product and the convolution product.

Prime ideals in infinite products of commutative rings

We describe the prime ideals and, in particular, the maximal ideals in products [Formula: see text] of families [Formula: see text] of commutative rings. We show that every maximal ideal is induced by an ultrafilter on the Boolean algebra [Formula: see text], where [Formula: see text] is the spectrum of maximal ideals of [Formula: see text], and [Formula: see text] denotes the power set. If every [Formula: see text] is in a certain class of rings including finite character domains and one-dimensional domains, we completely characterize the maximal ideals of [Formula: see text]. If every [Formula: see text] is a Prüfer domain, we completely characterize all prime ideals of [Formula: see text].

Projectively coresolved Gorenstein flat dimension of groups

In this paper, we introduce and study the projectively coresolved Gorenstein flat dimension of a group G over a commutative ring R and we prove that this dimension enjoys all the properties of the cohomological and the Gorenstein cohomological dimension. We also provide good estimations for the Gorenstein global dimension of RG in terms of this dimension and the Gorenstein global dimension of R. Moreover, we study special cases of groups, such as ▪-groups, and show that for such a group every Gorenstein projective RG-module is Gorenstein flat when the global dimension of R is finite.

Subrings of polynomial rings and the conjectures of Eisenbud and Evans

Let R be a commutative Noetherian ring of dimension d. In 1973, Eisenbud and Evans proposed three conjectures on the polynomial ring R[T]. These conjectures were settled in the affirmative by Sathaye, Mohan Kumar and Plumstead. One of the primary objectives of this article is to investigate the validity of these conjectures over Noetherian subrings of R[T] of dimension d+1, containing R. We formulate a class of such rings, which includes polynomial rings, Rees algebras, Rees-like algebras and Noetherian symbolic Rees algebras, and exhibit that all three conjectures hold for rings belonging to this class.

Gamma Graph Γ(Z_n)(γ) Of Zero Divisor Graph Of A Finite Commutative Ring Z_n

Consider the family of γ-sets of a zero-divisor graph of finite commutative ring and define the γgraphs (γ) = ( of to be the graph whose vertice V(γ) corresponds 1-to-1 with the γ-sets , say S1 and S2, form an edge in E(γ) if there exist a vertex vÎ such that (i)v is adjacent to and (ii) and Using this definition, we investigate the interplay between the graph theoretic properties of and and the ring theoretic properties of Further, we prove that are an Eulerian and Hamiltonian.

Noetherian spectrum condition and the ring 𝒜[X

Let [Formula: see text] be an ascending chain of commutative rings with identity and let [Formula: see text] be the ring of polynomials with coefficient of degree [Formula: see text] in [Formula: see text] for each [Formula: see text] In the first part of this paper, we give a necessary and sufficient conditions for the ring [Formula: see text] to satisfy Noetherian spectrum (a ring [Formula: see text] is said to have Noetherian spectrum if [Formula: see text] satisfies the Ascending Chain Condition (ACC) on radical ideals). In particular, we extend classical results proved by Ohm and Pendleton in [J. Ohm and R. Pendleton, Rings with Noetherian spectrum, Duke Math. J. 35 (1968) 631–639] and by Hizem in [S. Hizem, Chain conditions in rings of the form A + XB[X] and A + XI[X], in Commutative Algebra and its Applications (de Gruyter, 2009), pp. 259–274]. The second part of this paper is motivated by the interesting results proved on uniformly [Formula: see text]-Noetherian rings by Kim et al. in [W. Qi, H. Kim, F. Wang, M. Chen and W. Zhao, Uniformly S-Noetherian rings (submitted)]. We introduce the concept of uniformly [Formula: see text]-Noetherian spectrum and study its properties. Let [Formula: see text] be a commutative ring and [Formula: see text] a multiplicative subset of [Formula: see text] We say that [Formula: see text] has uniformly[Formula: see text]-Noetherian spectrum if that there exists an [Formula: see text] such that for any ideal [Formula: see text] of [Formula: see text], [Formula: see text] for some finitely generated subideal [Formula: see text] of [Formula: see text] We give the Eakin–Nagata–Formanek theorem for the uniformly [Formula: see text]-Noetherian spectrum condition. We also study the uniformly [Formula: see text]-Noetherian uniformly properties on several ring constructions (Nagata’s idealization and amalgamated algebras).

Mass Formula for Self-Orthogonal and Self-Dual Codes over Non-Unital Rings of Order Four

We study the structure of self-orthogonal and self-dual codes over two non-unital rings of order four, namely, the commutative ring I=a,b|2a=2b=0,a2=b,ab=0 and the noncommutative ring E=a,b|2a=2b=0,a2=a,b2=b,ab=a,ba=b. We use these structures to give mass formulas for self-orthogonal and self-dual codes over these two rings, that is, we give the formulas for the number of inequivalent self-orthogonal and self-dual codes, of a given type, over the said rings. Finally, using the mass formulas, we classify self-orthogonal and self-dual codes over each ring, for small lengths and types.

Motivic Pontryagin classes and hyperbolic orientations

AbstractWe introduce the notion of hyperbolic orientation of a motivic ring spectrum, which generalises the various existing notions of orientation (by the groups , , , ). We show that hyperbolic orientations of ‐periodic ring spectra correspond to theories of Pontryagin classes, much in the same way that ‐orientations of arbitrary ring spectra correspond to theories of Chern classes. We prove that ‐periodic hyperbolically oriented cohomology theories do not admit further characteristic classes for vector bundles, by computing the cohomology of the étale classifying space . Finally, we construct the universal hyperbolically oriented ‐periodic commutative motivic ring spectrum, an analogue of Voevodsky's cobordism spectrum .

Some applications of eigenvalues of unitary Cayley graphs of matrix rings over finite fields

ABSTRACT In this work, we provide two applications of the eigenvalues of the unitary Cayley graphs over matrix rings over finite fields. For a ring R, denotes the Jacobson radical of R. In 2012, Kiani and Aghaei conjectured that if R and S are finite rings and their unitary Cayley graphs are isomorphic, then and are isomorphic. If this conjecture holds and , then R is characterized by its unitary Cayley graph, and we say that R is a ring determined by unitary Cayley graphs. Kiani and Aghaei showed that every finite commutative ring and are such rings. We examine the eigenvalues of and obtain many new families of rings determined by unitary Cayley graphs. In 2021, Podestá and Videla characterized all finite commutative rings R such that the graphs in the triple are mutually equienergetic non-isospectral and Ramanujan where is the unitary Cayley sum graph. We use the eigenvalues of the graph and some observation on the number of matrices of the given rank to extend Podestá and Videla's results by working on the finite non-commutative ring being a product of matrix rings over local rings.

A topological property of a hypergraph assigned to commutative rings

Studying algebraic structures via graphs and hypergraphs assigned to them can be of interest. Especially, computing the genus of a graph as a topological index leads to a better understanding of the related algebraic structure. In this direction we apply a hypergraph, namely [Formula: see text]-zero divisor hypergraph assigned to a commutative ring and study its genus. In this paper, we characterize all finite commutative nonlocal rings [Formula: see text] with identity whose [Formula: see text] has genus two. Further, we classify all finite commutative nonlocal rings [Formula: see text] whose [Formula: see text] has crosscap two. Moreover, we provide a MATLAB code for calculating [Formula: see text]-zero-divisor of [Formula: see text] and the hyperedge of [Formula: see text]-zero-divisor hypergraph of [Formula: see text].

Ideals In Matrix Rings Over Commutative Rings

In this research, we discuss about ideal of matrix rings over commutative rings and its properties. The research of ideal in matrix rings is important because it is the basic structure for constructing factor rings in matrix rings. This research is a literature research that examines and develops research that has been done previously. We develop ideal concepts in an usually ring into matrix rings over commutative rings. By showing the sufficient and necessary condition of ideal of matrix rings over commutative rings, we show the form of ideal in matrix rings over commutative rings. Then, by using the properties of two ideal in a ring, we show the properties of intersection, addition and multiplication of two ideal in matrix rings over commutative rings. The result of this research is the form of ideal in matrix rings over commutative rings is the set of all matrices over the ideal in commutative rings. Then, the intersection, addition and multiplication of two ideal in matrix rings over commutative rings is also an ideal of matrix rings over commutative rings.

On the Goldie dimension of finitely generated locally cyclic modules

Let R be a commutative ring with identity. In this paper we investigate the Goldie dimension of finitely generated locally cyclic R-modules. Then, we give a characterization of rings whose finitely generated locally cyclics have finite Goldie dimension.