Let R be a commutative ring and let X,M be two unitary R-modules. In this note we introduce the notion of a prime submodule relative to a homomorphism ϕ ( ϕ -prime submodules). Several properties of ϕ -prime submodules are presented and we find two conditions under which ϕ -prime gives prime. Then its behavior with fractions formation on a multiplicatively closed subset S of R will be studied. When P is an invertible cyclic submodule of M, we characterize the primness of P in terms of ϕ -primness of P for all ϕ ∈ Hom R ( P − 1 , M ) , and for all ϕ ∈ Hom R ( End R ( P ) , M ) . Then, we consider maximal ϕ -prime ( ϕ -primary) submodules and see that maximal ϕ -primary and maximal ϕ -prime coincide. The results of this paper are initiated from the paper [A. Mimouni, On functional prime ideals in commutative rings, Comm. Alg. 52(11), 2024, 4525–4533]. Communicated by Scott Chapman

Over a commutative Noetherian ring, we show that the Auslander-Reiten conjecture holds true for the class of (finitely generated) modules whose dual has finite complete intersection dimension. We provide another result that validates the conjecture for the class of modules whose self-dual has finite complete intersection dimension and either the module or its dual has finite Gorenstein dimension. Thus we combine and strengthen a number of results in the literature, due to Auslander-Ding-Solberg, Dey-Ghosh and Rubio-Pérez.

Abstract Let be a Noetherian commutative ring and be a regular sequence in . We introduce a framework to study by linking the Koszul cohomology of on the sequence and local cohomology modules . As an application, we prove that if is a Noetherian regular ring of prime characteristic and form a regular sequence, then is Zariski‐closed for each integer and each ideal .

In this paper, we investigate generalized concatenated codes over finite commutative chain rings and derive two lower bounds on their distances with respect to an arbitrary weight function. One bound generalizes Jensen’s lower bound, while the other extends a recent bound obtained by Özbudak and Özkaya (2024), with the latter proving to be as effective as the former and, under certain conditions, even superior. We also provide a trace description for all quasi-abelian codes over finite commutative chain rings, offering a construction method for these codes. Using this description, we show that quasi-abelian codes can be viewed as generalized concatenated codes. Additionally, we derive the generalized concatenated structure of the Euclidean dual code of each quasi-abelian code. We also derive two lower bounds on the homogeneous distances of quasi-abelian codes over finite commutative chain rings, using their generalized concatenated structure. As applications, we demonstrate that quasi-abelian codes and their special class consisting of Euclidean LCD codes are asymptotically good with respect to the homogeneous metric, establishing the existence of two asymptotically good classes of linear codes over finite commutative chain rings.

This study introduces and investigates the idea of $S$-pm-rings, a generalization of pm-rings in the context of commutative rings with a multiplicatively closed subset $S$. We prove that a ring $R$ is an $S$-pm-ring if and only if its $S$-maximal spectrum is a retract (specifically, a deformation retract) of its $S$-prime spectrum. Furthermore, we establish the equivalence of the $S$-pm-ring property to the normality of the $S$-prime spectrum and the Hausdorff property of the $S$-maximal spectrum. We also explore the relationship between $S$-pm-rings and $S$-clean rings, demonstrating that every $S$-local ring is $S$-clean, and every $S$-clean ring is an $S$-pm-ring. These results extend classical theorems in commutative algebra and algebraic geometry to the $S$-version context.

In this paper, we introduce weakly classical 1-absorbing primary submodules of an unital module over a commutative ring with nonzero identity. Consider [Formula: see text] as a module over a ring [Formula: see text]. A proper submodule [Formula: see text] of [Formula: see text] is called weakly classical 1-absorbing primary submodule, if for each nonunit [Formula: see text] and [Formula: see text] with [Formula: see text], it follows that either [Formula: see text] or [Formula: see text] for some [Formula: see text]. Various properties and characterizations of weakly classical 1-absorbing primary submodules are explored.

Let $R$ be a commutative Noetherian ring, $I$ an ideal of $R$ and $M$ an arbitrary $R$-module. We show that if the set $X=\cup_{i\geq 2}{\rm Supp}_R({\rm H}^{i}_{I}(R))$ is finite, then the category of all weakly cofinite modules with respect to the ideal $I$ of $R$ is an Abelian subcategory of the category of $R$-modules. In particular, it is true whenever ${\rm q}(I,R)\leq 1$.

Abstract We develop the theory of weakly S -square-difference factor absorbing ideals (weakly S -sdf ideals) in a commutative ring R , extending the notion of weakly sdf-absorbing ideals to a relative setting determined by a multiplicatively closed subset $$S\subseteq R$$ S ⊆ R . We establish their basic properties, relate them to weakly S -prime, S -sdf-absorbing, and S -semiprime ideals, and use the S -characteristic and S -invertible elements to identify conditions under which weakly S -sdf ideals strengthen to weakly S -prime or S -sdf-absorbing ideals. We further examine their behavior under homomorphic images, direct products, quotients, trivial ring extensions, amalgamated duplications, and amalgamated algebras. A complete characterization of weakly S -sdf ideals in pullback rings $$A\times _{C}B$$ A × C B is obtained, showing that the property transfers precisely through the coordinate ideals together with a natural square-difference compatibility condition.

The unit graph of a commutative ring with a non-zero identity is a graph with vertices as ring elements, and there is an edge between two distinct vertices if their sum is a unit. This study investigates the decomposition of the unit graph by examining its induced subgraphs and analyze key graph invariants, such as connectivity, diameter, and girth, for a finite local ring. We further decompose the unit graph of certain finite commutative rings into fundamental structures, such as cycle and star graphs.

On the eigenvalues of zero divisor graphs associated with commutative rings

We study commutative local rings over which every upper-triangular matrix is the sum of an idempotent and a $q$-potent that commute. For Galois rings and rings of the form $\mathbb{F}_{p^{k}}[x]/\langle x^{r} \rangle$, necessary and sufficient criterion are provided.

Let R be a 2-torsion-free commutative ring with unity, X a locally finite preordered set, and I ( X , R ) the incidence algebra of X over R with center Z ( I ( X , R ) ) . In this paper, we provide a condition under which every nonlinear Lie n-centralizer θ on I ( X , R ) is proper. Under the same condition, we also show that every nonlinear generalized Lie n-derivation L of I ( X , R ) can be written as L ( f ) = λ f + D ( f ) , ∀ f ∈ I ( X , R ) , where λ ∈ Z ( I ( X , R ) ) and D is a nonlinear Lie n-derivation.

Construction of a linear code over a finite commutative ring generated by a set of minimal codewords

The product matrix of a finite commutative ring R = { x 1 , x 2 , … , x n } and an element u ∈ R is the matrix A u ( R ) = [ a ij ] , where a ij = 1 if x i x j = u , and a ij = 0 otherwise. This provides a natural extension of the concept of the adjacency matrix of the zero-divisor graph of a ring, which has been studied extensively. In this paper, we find the characteristic polynomial of A u ( R ) for a local ring R of odd order and a unit u. By studying the structure of a finite local ring, we find the characteristic polynomial of A u ( R ) for a local ring R and any u ∈ R in two cases: when the Jacobson radical of R has either the maximal or the minimal possible index of nilpotency.

For a graph with edges and vertices, the general zeroth-order Randić index measures the sum of the degree of each vertex to the power of nonzero ω. Meanwhile, the zero divisor graph of a commutative ring R is the set of all zero divisors in R in which two vertices are adjacent if their product is zero. In this paper, the Zagreb indices of the zero divisor graph for the commutative ring Z p k q are found for the cases ω = 1 , 2 , and 3 where k is a positive integer, p and q are primes with p < q .

This paper studies residually finite rings, a class of commutative rings in which every nonzero ideal has a finite quotient. We begin by examining several properties of residually finite rings, with particular emphasis on how residual finiteness is preserved under distinguished ring extensions, including direct products, flat overrings, finite extensions, polynomial and power series rings, as well as Nagata and Anderson rings. We also provide a negative answer to an open question posed by Ion and Militaru. Finally, we introduce and investigate a proper generalization of this concept, which we call locally residually finite rings.

On normal subgroups of twisted Chevalley groups over commutative rings

Recursive construction and enumeration of self-orthogonal and self-dual codes over finite commutative chain rings of even characteristic

As a generalization of vector spaces over finite fields, we study vector spaces over finite commutative rings, and obtain Anzahl formulas and a dimensional formula for subspaces. By using these results, we discuss normalized matching (NM) property of a class of subspace posets.

Let Z(R)′ denote the set of all elements in the ring R that are neither zero nor units, where R is assumed to be a commutative ring with a multiplicative identity satisfying 1≠0. Two distinct vertices w and κ are defined to be adjacent if and only if κ does not lie in the ideal generated by w in R, that is, κ∉wR, and simultaneously, w does not lie in the ideal generated by κ in R, that is, w∉κR. The cozero-divisor graph of R, denoted by Γ′(R), is an undirected graph in which the vertices are given by the set Z(R)′. This article presents a comprehensive evaluation of both the Euler Sombor index and the average Sombor index for the graphs Γ′(Zn) corresponding to various values of n.