Morita Equivalence and Homological Characterization of Semirings

Abstract

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 Universalis 51 (2004) [287][288][289][290][291][292][293][294][295][296][297][298][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. 445J. Algebra Appl. 2011.10:445-473. Downloaded from www.worldscientific.com by UNIVERSITY OF AUCKLAND LIBRARY -SERIALS UNIT on 03/15/15. For personal use only. 446 Y. Katsov & T. G. Nam studying semirings, and their representations/semimodules, perhaps explains why the research on categorical and homological aspects of theory of semirings and semimodules is still behind that for rings and monoids. However, as have been emphasized in number of recent publications (see, for instance, a recent monograph [28; 31, Question 9.5.20]), in many areas of modern mathematics -in particular, in such booming areas as idempotent and tropical algebra, algebraic geometry and mathematics -there have been expressed a strong interest in, and even need for, developing of categorical and homological methods of theory of semirings and semimodules. In this connection, one of the main goals of this paper is to narrow this "deficit" and promote, and further develop the fundamental categorical and homological methods, originated in [19,21,22], and then considered in [15,16,23], in the semiring setting, as well as to build a solid ground on, and applications of, these methods for future research on semirings and semimodules.In the same time in the modern homological theory of modules over rings, the results characterizing rings by properties of modules and/or suitable categories of modules over them are of great importance and sustained interest (for a good number of such results one may consult, for example, [2, 26; see also 33]). Inspired by this, during the last three decades quite a few results related to this genre have been obtained in different non-additive settings. Just to mention some of these settings, we note that a very valuable collection of numerous interesting results on characterizations of monoids by properties and/or by categories of acts over them, i.e. on so-called homological classification of monoids, can be found in [24]; and, for the results on "homological classification of distributive lattices," one may consult the survey [8]. In light of this, a presentation of some of the applications of the categorical and homological considerations and technique, developed in the first part of the paper, to problems of homological characterization/classification of semiring...

FP-injective semirings, semigroup rings and Leavitt path algebras

“…By [14,Proposition 4.8] and [14,Lemma 4.10], we obtain that M ∈ | T M| is finitely generated and projective if and only if F (M ) ∈ | S M| is finitely generated and projective, respectively. Now, let M be an FP-injective left T -semimodule.…”

“…where G(i) is an injective T -homomorphism, by the dual of [14,Lemma 4.7]. By the above observation, we may consider G(X) to be a finitely generated subsemimodule of the finitely generated projective left T -semimodule G(S n ).…”

Toward homological characterization of semirings by e-injective semimodules

“…Since projective semimodules are e-flat by Lemma 6.8, both F and G preserve short exact sequences, and the statement readily follows from the natural functor isomorphisms F G ≃ Id S M and GF ≃ Id T M [38,Sect. 4.4] (see also the dual of [26,Lemma 4.10]). …”

“…Case II: Let S ≃ End(L) for some nonzero finite distributive lattice L. Then, by [28,Theorem 5.7], S is a simple semiring that Morita equivalent to the Boolean semiring B; and let the functors F : S M ⇄ B M : G establish an equivalence between the semimodule categories S M and B M. For any finitely generated left S-semimodule M ∈ | S M|, by [26,Proposition 4.8] and Theorem 4.4 (5), the semimodule F (M) ∈ | B M| is a finitely generated e-injective B-semimodule. Then, applying Lemma 6.9 and the natural isomorphism M ≃ G(F (M)), we have that the semimodule M ∈ | S M| is e-injective as well and end the proof.…”

“…As algebraic objects, semirings are certainly the most natural generalization of such (at first glance different) algebraic systems as rings and bounded distributive lattices, and therefore, they form an extremely interesting, natural, and important, non-abelian/non-additive setting for furthering of the homological characterization, i.e., characterizing semirings by properties of suitable classes (categories) of semimodules over them. In fact, this is an ongoing project of an substantial interest (see, e.g., [22], [24], [23], [16], [25], [17], [26], [18], [19], and [3]). In all studies regarding the homological characterization, the concepts of 'injectivity' and 'projectivity' of objects -R-modules, S-acts, S-semimodules, etc.-play the most leading role.…”