Journal of Algebra and Its Applications

The first aim of this article is to study maximal ideals of a preadditive category . Maximal ideals, which do not exist in general for arbitrary preadditive categories, are associated to a maximal ideal of the endomorphism ring of an object and always exist when the category is semilocal. If is additive and semilocal, any skeleton of is a Krull monoid and we are able to characterize the essential valuations of and provide some natural divisor homomorphisms and divisor theories of .

John Dauns died on June 4, 2009 of cancer in New Orleans at the age of 73. His work on rings and modules is well-known in the algebra community. However, functional analysts working in the area of C*-algebras are likely to know his name from one theorem that is a corollary of results he and I had obtained in work we did in the mid-sixties of the last century ([17, 27, 18]) and which became known as the Dauns-Hofmann Theorem in C*-algebra theory. It has been known in the C*-text book and monograph literature up to the recent one under this name ([9, 20, 21, 39, 41]) since it provides apparently a useful tool that continues to be used in current research (see e.g. [7, 8, 25, 38, 40]). The problem with the historical record of the Dauns–Hofmann Theorem is that it used to be somewhat obscure how it originated and that the full weight of what was proved was not precisely understood for a long time. As John Dauns was deeply, if not subbornly involved in the development of the early phases of the representation of rings, algebras, C*-algebras (and other classes of algebras) by continuous sections in bundles (sometimes called continuous fields) [11–18], and since his work in this area was substantial and contributed much to a local culture of "sectional representation" at Tulane University ([19, 22, 27–34, 43, 44, 48]), I feel that it is justified to attempt a clarification. He can no longer participate himself in such an attempt; nor would he actually protest the occasional lack of acknowledgment were he alive, because that would be contrary to his ever gentle disposition. This small survey is devoted to shedding some additional light on this portion of John Dauns' work in mathematics; it is natural that it should have a personal tenor by a writer remembering his presence and his work as a collaborator.

Let B be a torsion-free finitely faithful S-group. This paper investigates the class of all torsion-free Abelian groups A with the property that Ext(A, B) is torsion-free. Our discussion is based on a natural extension of the concept of coseparability.

In a matrix ring R = 𝕄₂(S) where S is a commutative ring, we study equations of the form XY - YX = U ∈ GL₂(S), focusing on matrices in R that can appear as X or as XY in such equations. These are the completable and the reflectable matrices in R. For matrices A ∈ R with a zero row or with a constant diagonal, explicit and "computer-checkable" criteria are found for A to be completable or reflectable. A formula for det (XY - YX) discovered recently with Shomron connects this study to diophantine questions about the representation of units of the ground ring S by quadratic forms of the type px² + qy².

Let R be a ring with identity. A unital left R-module M has the min-property provided the simple submodules of M are independent. On the other hand a left R-module M has the complete max-property provided the maximal submodules of M are completely coindependent, in other words every maximal submodule of M does not contain the intersection of the other maximal submodules of M. A semisimple module X has the min-property if and only if X does not contain distinct isomorphic simple submodules and this occurs if and only if X has the complete max-property. A left R-module M has the max-property if for every positive integer n and distinct maximal submodules L, L_i (1 ≤ i ≤ n) of M. It is proved that a left R-module M has the complete max-property if and only if M has the max-property and every maximal submodule of M/Rad M is a direct summand, where Rad M denotes the radical of M, and in this case every maximal submodule of M is fully invariant. Various characterizations are given for when a module M has the max-property and when M has the complete max-property.

Any finite set of linear operators on an algebra A yields an operator algebra B and a module structure on A, whose endomorphism ring is isomorphic to a subring A^B of certain invariant elements of A. We show that if A is a critically compressible left B-module, then the dimension of its self-injective hull  over the ring of fractions of A^B is bounded by the uniform dimension of A and the number of linear operators generating B. This extends a known result on irreducible Hopf actions and applies in particular to weak Hopf action. Furthermore we prove necessary and sufficient conditions for an algebra A to be critically compressible in the case of group actions, group gradings and Lie actions.

Let M and N be two modules over a ring R. The concern is about the four substructures of hom_R(M, N): the Jacobson radical J[M, N], the singular ideal Δ[M, N], the co-singular ideal ∇[M, N] and the total Tot[M, N]. One natural question is to characterize when the total is equal to one or more of the other structures. We review some known results and prove several new results towards this question and, as consequences, give answers to some related questions.

Any (co)ring R is an endofunctor with (co)multiplication on the category of abelian groups. These notions were generalized to monads and comonads on arbitrary categories. Starting around 1970 with papers by Beck, Barr and others a rich theory of the interplay between such endofunctors was elaborated based on distributive laws between them and Applegate's lifting theorem of functors between categories to related (co)module categories. Curiously enough some of these results were not noticed by researchers in module theory and thus notions like entwining structures and smash products between algebras and coalgebras were introduced (in the nineties) without being aware that these are special cases of the more general theory.

The purpose of this survey is to explain several of these notions and recent results from general category theory in the language of elementary module theory focusing on functors between module categories given by tensoring with a bimodule. This provides a simple and systematic approach to smash products, wreath products, corings and rings over corings (C-rings). We also highlight the relevance of the Yang–Baxter equation for the structures on the threefold tensor product of algebras or coalgebras (see 3.6).

The quasi-Baer condition of R/P(R) is investigated when R is a quasi-Baer ring, where P(R) is the prime radical of R. We provide an example of quasi-Baer ring R such that R/P(R) is not quasi-Baer. However, when P(R) is nilpotent, we prove that if R is a quasi-Baer (resp., Baer) ring, then R/P(R) is quasi-Baer (resp., Baer). Examples which illustrate and delimit the results of this paper are provided.

We consider modules over commutative rings. We are looking for sufficient conditions for a continuous well-ordered ascending chain of projective modules that are pure in their union to have a projective union. First, we deal with countable chains without restriction on the sizes of the modules; in this case no extra hypothesis is needed provided R is an integral domain or ℵ₀-noetherian (Theorem 1.3); in the latter case, the theorem covers pure-projectvity as well. We also consider chains of longer lengths over an integral domain. We establish a condition (in terms of a slightly modified κ⁺-freeness) for the projectivity of the union provided that the links of the chain are ≤ κ-generated and the length of the chain is κ⁺ for an infinite cardinal κ (Theorem 1.4). Our theorems generalize several results in the literature.