Topological characterizations of amenability and congeniality of bases

[EN] We provide topological interpretations of the recently introduced notions of amenability and congeniality of bases of innite dimensional algebras. In order not to restrict our attention only to the countable dimension case, the uniformity of the topologies involved is analyzed and therefore the pertinent ideas about uniform topological spaces are surveyed.A basis B over an innite dimensional F-algebra A is called amenable if FB, the direct product indexed by B of copies of the eld F, can be made into an A-module in a natural way. (Mutual) congeniality is a relation that serves to identify cases when different amenable bases yield isomorphic A-modules.(Not necessarily mutual) congeniality between amenable bases yields an epimorphism of the modules they induce. We prove that this epimorphism is one-to-one only if the congeniality is mutual, thus establishing a precise distinction between the two notions.López-Permouth, SR.; Stanley, B. (2020). Topological characterizations of amenability and congeniality of bases. Applied General Topology. 21(1):1-15. https://doi.org/10.4995/agt.2020.11488OJS115211L. M. Al-Essa, S. R. López-Permouth and N. M. Muthana, Modules over infinite-dimensional algebras, Linear and Multilinear Algebra 66 (2018), 488-496. https://doi.org/10.1080/03081087.2017.1301365P. Aydogdu, S. R. López-Permouth and R. Muhammad, Infinite-dimensional algebras without simple bases, Linear and Multilinear Algebra, to appear.J. Díaz Boils, S. R. López-Permouth and R. Muhammad, Amenable and simple bases of tensor products of infinite dimensional algebras, preprint.R. Engelking, General Topology, Sigma Series in Pure Mathematics, vol. 6 (1989).S. R. López-Permouth and B. Stanley, On the amenability profile of an infinite dimensional module over an algebra, preprint.P. Nielsen, Row and column finite matrices, Proc. Amer. Math. Soc. 135, no. 9 (2007), 2689-2697. https://doi.org/10.1090/S0002-9939-07-08790-4B. Stanley, Perspectives on amenability and congeniality of bases, Ph. Dissertation, Ohio University, February 2019.S. Willard, General Topology, Dover Publications (1970)

On the equivalence of codes over rings and modules

AbstractIn light of the result by Wood that codes over every finite Frobenius ring satisfy a version of the MacWilliams equivalence theorem, a proof for the converse is considered. A strategy is proposed that would reduce the question to problems dealing only with matrices over finite fields. Using this strategy, it is shown, among other things, that any left MacWilliams basic ring is Frobenius. The results and techniques in the paper also apply to related problems dealing with codes over modules

Rational power series, sequential codes and periodicity of sequences

AbstractLet R be a commutative ring. A power series f∈R[[x]] with (eventually) periodic coefficients is rational. We show that the converse holds if and only if R is an integral extension over Zm for some positive integer m. Let F be a field. We prove the equivalence between two versions of rationality in F[[x1,…,xn]]. We extend Kronecker’s criterion for rationality in F[[x]] to F[[x1,…,xn]]. We introduce the notion of sequential code which is a natural generalization of cyclic and even constacyclic codes over a (not necessarily finite) field. A truncation of a cyclic code over F is both left and right sequential (bisequential). We prove that the converse holds if and only if F is algebraic over Fp for some prime p. Finally, we show that all sequential codes are obtained by a simple and explicit construction

An alternative perspective on projectivity of modules

Similar to the idea of relative projectivity, we introduce the notion of relative subprojectivity, which is an alternative way to measure the projectivity of a module. Given modules $M$ and $N$, $M$ is said to be {\em $N$-subprojective} if for every epimorphism $g:B \rightarrow N$ and homomorphism $f:M \rightarrow N$, then there exists a homomorphism $h:M \rightarrow B$ such that $gh=f$. For a module $M$, the {\em subprojectivity domain of $M$} is defined to be the collection of all modules $N$ such that $M$ is $N$-subprojective. A module is projective if and only if its subprojectivity domain consists of all modules. Opposite to this idea, a module $M$ is said to be {\em subprojectively poor}, or {\em $sp$-poor} if its subprojectivity domain is as small as conceivably possible, that is, consisting of exactly the projective modules. Properties of subprojectivity domains and $sp$-poor modules are studied. In particular, the existence of an $sp$-poor module is attained for artinian serial rings.Comment: Dedicated to the memory of Francisco Raggi; v2 some editorial changes. 'Right hereditary right perfect' replaced by the (equivalent) condition 'right hereditary semiprimary'; v3 a mistake corrected in the statements of Propositions 3.8 and 3.

Characterizing rings in terms of the extent of injectivity and projectivity of their modules

Given a ring R, we define its right i-profile (resp. right p-profile) to be the collection of injectivity domains (resp. projectivity domains) of its right R-modules. We study the lattice theoretic properties of these profiles and consider ways in which properties of the profiles may determine the structure of rings and viceversa. We show that the i-profile is isomorphic to an interval of the lattice of linear filters of right ideals of R, and is therefore modular and coatomic. In particular, we give a practical characterization of the i-profile of a right artinian ring. We show through an example that the p-profile is not necessarily a set, and also characterize the right p-profile of a right perfect ring. The study of rings in terms of their (i- or p-)profile was inspired by the study of rings with no (i- or p-) middle class, initiated in recent papers by Er, L\'opez-Permouth and S\"okmez, and by Holston, L\'opez-Permouth and Orhan-Ertas. In this paper, we obtain further results about these rings and we also use our results to provide a characterization of a special class of QF-rings in which the injectivity and projectivity domains of any module coincide.Comment: 19 pages, examples and propositions added. Title change

Invertible Algebras with an augmentation map

Abstract. We briefly survey results about Invertible Algebras (algebras having bases that consist entirely of units) and other related notions. In addition, we consider the existence of an augmentation map as a possible way in which results about group rings, the archetypical invertible algebras, may be extended to more general settings. Invertible Algebras and Some Related Notions Section 1 of this paper surveys the study of Invertible Algebras (over not-necessarily commutative rings) and other related notions. Section 2 introduces new ideas analogous to the augmentation map of group rings. Invertible algebras are those algebras that satisfy the condition that they have a basis consisting entirely of units. Our brief survey also touches on a few other related concepts. The concept of invertible algebras was originally introduced in [17]) and S-rings (c.f. [15]). Group Rings are clearly an example of invertible algebras; their theory is very well developed and is central in many areas of mathematics. Standard references for group rings include the classics In this paper, when we use the expression A is an R-algebra we deviate from the standard use of that terminology in two ways: one which narrows the net that we cast and another one that widens it. First, we do not allow a proper homomorphic image of the ring R to be contained in A; according to the definition we use in this paper, R itself is contained in A. The second difference is that R is not necessarily assumed to be contained in the center of A; in fact, we do not even assume that the ring R is commutative. Also, a feature that will be common to all those algebras considered here is that they will be free as (left-) R-modules. In other words, our setting is that we have a ring A that has a subring R such that A is a free left R-module. Definition 1.1. Let A be an algebra over a ring R and B be a basis for A over R. B is an invertible basis if each element of B is invertible in A. If B is an invertible basis such that B −1 , the set of the inverses of the elements of B, also constitutes a basis then B is an invertible-2(I2) basis. An algebra with an invertible basis is an invertible algebra and an algebra with an I2 basis is an I2 algebra. Various papers in the literature have considered properties of rings having to do with expressing their elements in terms of sums of units. See ([4] (b) If R is an S-ring that contains a unit u such that u + 1 is a unit, then R is an even S-ring

Advances in rings and modules

This volume, dedicated to Bruno J. Müller, a renowned algebraist, is a collection of papers that provide a snapshot of the diversity of themes and applications that interest algebraists today. The papers highlight the latest progress in ring and module research and present work done on the frontiers of the topics discussed. In addition, selected expository articles are included to give algebraists and other mathematicians, including graduate students, an accessible introduction to areas that may be outside their own expertise