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...

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In this paper, among other results, there are described (complete) simple -simultaneously ideal-and congruence-simple -endomorphism semirings of (complete) idempotent commutative monoids; it is shown that the concepts of simpleness, congruence-simpleness and ideal-simpleness for (complete) endomorphism semirings of projective semilattices (projective complete lattices) in the category of semilattices coincide iff those semilattices are finite distributive lattices; there are described congruence-simple complete hemirings and left artinian congruence-simple complete hemirings. Considering the relationship between the concepts of 'Morita equivalence' and 'simpleness' in the semiring setting, we have obtained the following results: The ideal-simpleness, congruencesimpleness and simpleness of semirings are Morita invariant properties; A complete description of simple semirings containing the infinite element; The representation theorem -"Double Centralizer Property" -for simple semirings; A complete description of simple semirings containing a projective minimal onesided ideal; A characterization of ideal-simple semirings having either infinite elements or a projective minimal one-sided ideal; A confirmation of Conjecture of [18] and solving Problem 3.9 of [17] in the classes of simple semirings containing either infinite elements or projective minimal left (right) ideals, showing, respectively, that semirings of those classes are not perfect and the concepts of 'mono-flatness' and 'flatness' for semimodules over semirings of those classes are the same. Finally, we give a complete description of ideal-simple, artinian additively idempotent chain semirings, as well as of congruence-simple, latticeordered semirings.

Among other results on subtractive semimodules and semirings, we present various (homological) characterizations of subtractive semisimple semirings. Also, we give complete descriptions of finite subtractive semisimple as well as additively regular (in particular, additively idempotent) subtractive semisimple semirings.