Journal of Algebra and Its Applications

Let Λ be a commutative local uniserial ring of length n, p be a generator of the maximal ideal, and k be the radical factor field. The pairs (B, A) where B is a finitely generated Λ-module and A ⊆B a submodule of B such that p^mA = 0 form the objects in the category . We show that in case m = 2 the categories are in fact quite similar to each other: If also Δ is a commutative local uniserial ring of length n and with radical factor field k, then the categories and are equivalent for certain nilpotent categorical ideals and . As an application, we recover the known classification of all pairs (B, A) where B is a finitely generated abelian group and A ⊆ B a subgroup of B which is p²-bounded for a given prime number p.

We consider an affine superscheme X over a field K, of characteristic that is different from 2, together with an action of a finite supergroup G on it. We prove that if G acts freely on X, then the sheaf quotient (with respect to fppf topology) is again an affine superscheme Y, where K[Y] ≃ K[X]^G. Besides, K[X] is a finitely presented progenerator as a K[X]^G-module.

In this paper, we define and study a special type of trisections in a module category, namely the compact trisections which characterize quasi-directed components. We apply this notion to the study of laura algebras and we use it to define a class of algebras with predictable Auslander–Reiten components.

We give a characterization of the prime natural numbers p of the form x^q + ry^q, where q and r are distinct odd primes.

In this paper, we consider from two different perspectives the concept of "Morita equivalence" in a (non-additive) semiring setting, as well as its application to homological characterization of semirings. Among other results, we present an analog of the Eilenberg–Watts theorem for module categories in the semimodule setting and give various homological characterizations of semisimple and subtractive semirings. We also solve Problem 3.9 in [Y. Katsov, On flat semimodules over semirings, Algebra Universalis51 (2004) 287–299] for the class of additively regular semisimple semirings, showing that for semimodules over semirings of this class the concepts of "mono-flatness" and "flatness" coincide.

Let R be a ring. Modules satisfying ascending or descending chain conditions (respectively, acc and dcc) on non-summand submodules belongs to some particular classes , such as the class of all R-modules, finitely generated, finite-dimensional and cyclic modules, are considered. It is proved that a module M satisfies acc (respectively, dcc) on non-summands if and only if M is semisimple or Noetherian (respectively, Artinian). Over a right Noetherian ring R, a right R-module M satisfies acc on finitely generated non-summands if and only if M satisfies acc on non-summands; a right R-module M satisfies dcc on finitely generated non-summands if and only if M is locally Artinian. Moreover, if a ring R satisfies dcc on cyclic non-summand right ideals, then R is a semiregular ring such that the Jacobson radical J is left t-nilpotent.

Let A be a commutative integral domain with quotient field L, and let R be a subdomain of A with quotient field K. Assuming that L is a Galois extension of K, Nagata required the condition for R to be normal when A is called a Galois extension of R (see p. 31, M. Nagata, Local Rings (Wiley, New York, 1962)). However in this paper, A is considered in the case that R is not necessarily assumed to be normal. We introduce the notion of cyclic Galois extensions of integral domains and investigate several properties of such ring extensions. In particular, we completely determine the seminormalization of A in an overdomain B such that both A ⊆B are cyclic Galois extensions of R.

Let _SM_R be an (S, R)-bimodule. We denote D_S(N) = {s ∈ S |sM ⊆ N} for any submodule N of M_R. We investigate the condition D_S(K + T) = D_S(K) + D_S(T) for submodules K and T of M_R. Various characterizations and properties are derived for modules with this property. In particular, for S = End(M_R), the π-projective modules are those modules M_R for which S = D_S(K) + D_S(T) whenever M = K+T, and we give some new results on these modules and lifting modules.

Let R be a non-commutative ring. The commuting graph of R denoted by Γ(R), is a graph with vertex set R\Z(R) and two vertices a and b are adjacent if ab = ba. It has been shown that the diameter of Γ(R)^c is less than 3. For a finite ring R we show that the diameter of Γ(R)^c is one if and only if R is the non-commutative ring on 4 elements. Also we characterize all rings where the complements of their commuting graphs are planar. Moreover, we identify the commuting graphs of rings of order pⁱ for i = 2, 3 and prime number p.

A ring R is called right ℵ₀-injective if every homomorphism from a countably generated right ideal of R to R_R can be extended to a homomorphism from R_R to R_R. In this note, some characterizations of ℵ₀-injective rings are given. It is proved that if R is semilocal, then R is right ℵ₀-injective if and only if every homomorphism from a countably generated small right ideal of R to R_R can be extended to one from R_R to R_R. It is also shown that if R is right noetherian and left ℵ₀-injective, then R is quasi-Frobenius. This result can be considered as an approach to the Faith–Menal conjecture.

In the author's work [S. A. Annin, Attached primes over noncommutative rings, J. Pure Appl. Algebra212 (2008) 510–521], a theory of attached prime ideals in noncommutative rings was developed as a natural generalization of the classical notions of attached primes and secondary representations that were first introduced in 1973 as a dual theory to the associated primes and primary decomposition in commutative algebra (see [I. G. Macdonald, Secondary representation of modules over a commutative ring, Sympos. Math.11 (1973) 23–43]). Associated primes over noncommutative rings have been thoroughly studied and developed for a variety of applications, including skew polynomial rings: see [S. A. Annin, Associated primes over skew polynomial rings, Commun. Algebra30(5) (2002) 2511–2528; and S. A. Annin, Associated primes over Ore extension rings, J. Algebra Appl.3(2) (2004) 193–205]. Motivated by this background, the present article addresses the behavior of the attached prime ideals of inverse polynomial modules over skew polynomial rings. The goal is to determine the attached primes of an inverse polynomial module M[x^-1] over a skew polynomial ring R[x;σ] in terms of the attached primes of the base module M_R. This study was completed in the commutative setting for the class of representable modules in [L. Melkersson, Content and inverse polynomials on artinian modules, Commun. Algebra26(4) (1998) 1141–1145], and the generalization to noncommutative rings turns out to be quite non-trivial in that one must either work with a Bass module M_R or a right perfect ring R in order to achieve the desired statement even when no twist is present in the polynomial ring "Let M_R be a module over any ring R. If M[x^-1]_R is a completely σ-compatible Bass module, then Att(M[x^-1]_S) = {𝔭[x] : 𝔭 ∈ Att(M_R)}." The sharpness of the results are illustrated through the use of several illuminating examples.

We study integrality of the Frobenius–Perron eigenvalues of the Cartan matrices for the principal blocks of some finite groups of Lie type with noncyclic abelian Sylow p-subgroups.

Let R be a commutative ring with identity admitting at least two nonzero zero-divisors. First, in this article we determine when the complement of the zero-divisor graph of R is connected and also determine its diameter when it is connected. Second, in this article we study the relationship between the connectedness of the complement of the zero-divisor graph of R to that of the connectedness of the complement of the zero-divisor graph of T where either T = R[x] or T = R[[x]] and we study the relationship between their diameters in the case when both the graphs are connected. Finally, we give some examples to illustrate some of the results proved in this article.