Morita invariants of semirings-II
01 Feb 2018-Asian-european Journal of Mathematics (World Scientific Publishing Company)-Vol. 11, Iss: 01, pp 1850014
TL;DR: In this paper, the counterparts of most of the results on Morita invariants of semirings for semimodules are investigated and the lattice isomorphisms between the s...
Abstract: The purpose of this paper is to investigate the counterparts of most of our results on Morita invariants of semirings for semimodules. Among others, we obtain the lattice isomorphisms between the s...
Citations
More filters
TL;DR: In this paper, the main purpose of the paper is to consider two Morita equivalent semirings R and S via Morita context instead of considering them via the equivalence of the resulting semimodule.
Abstract: The main purpose of the paper is to consider two Morita equivalent semirings R and S via Morita context 〈R,S,RPS,SQR,𝜃,ϕ〉 instead of considering them via the equivalence of the resulting semimodule...
2 citations
01 Jan 2018
TL;DR: In this paper, the existence of a Morita context with unitary bisemimodules and surjective maps implies that the two semirings involved have isomorphic quantales of ideals and lattices of congruences.
Abstract: We consider Morita contexts for semirings that have certain local units but not necessarily an identity element. We show that the existence of a Morita context with unitary bisemimodules and surjective maps implies that the two semirings involved have isomorphic quantales of ideals and lattices of congruences.
TL;DR: In this paper, the notions of k-regularity and intra-kregularity in bi-semimodules of a Morita context of semirings have been introduced.
Abstract: The notions of k-regularity and intra k-regularity in bi-semimodules of a Morita context of semirings have been introduced. Some characterizations of k-regular and intra kregular idempotent bi-semimodules of a Morita context of semirings have been obtained in terms of k-b-subsemimodules. Then the study of the interplay among various components of a Morita context of semirings in terms of k-regularity and intra k-regularity has been accomplished. Finally it has been shown that k-regularity and intra k-regularity are Morita invariant properties.
10 Apr 2021
TL;DR: In this paper, the necessary and sufficient conditions for two semirings with local units to be Morita equivalent were studied and some properties which remain invariant under Morita equivalence were obtained.
Abstract: In this paper we study some necessary and sufficient conditions for two semirings with local units to be Morita equivalent. Then we obtain some properties which remain invariant under Morita equivalence.
Cites background or result from "Morita invariants of semirings-II"
...The results obtained here are nothing but counterparts of the results of [17] in the setting of semirings with local units....
[...]
...2 [17] by replacing the identity by a local unit the rest of the proof can be completed....
[...]
...[11], Sardar and Gupta [16, 17] independently studied some properties of semirings...
[...]
...Later Katsov et al. [11], Sardar and Gupta [16, 17] independently studied some properties of semirings The first author is grateful to CSIR, Govt. of India, for providing research support....
[...]
References
More filters
TL;DR: In this article, the concept of "Morita equivalence" in a non-additive semiring setting, as well as its application to homological characterization of semirings are considered.
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 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.
54 citations
"Morita invariants of semirings-II" refers background in this paper
...Two semirings R and S are said to be Morita equivalent [6] if there exists a progenerator RP ∈ |RM| for RM such that S ∼= End(RP ) as semirings; equivalently two semirings R and S are Morita equivalent if the categories RM and SM are equivalent categories....
[...]
...Introduction and Preliminaries In the year 2011 Katsov and Nam [6] transferred the ring theoretic approach of Morita equivalence (see [1]) to semirings heavily using the notion of tensor product [5] for semimodules....
[...]
TL;DR: In this paper, the authors investigated various classes of semirings and complete semiirings regarding the property of being ideal-simple, congruence-simple or both, and showed that the concepts of simpleness, simpleness and simpleness are Morita invariant properties.
Abstract: In this paper, we investigate various classes of semirings and complete semirings regarding the property of being ideal-simple, congruence-simple, or both. Among other results, we describe (complete) simple, i.e. simultaneously ideal- and congruence-simple, endomorphism semirings of (complete) idempotent commutative monoids; we show 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; we also describe 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 obtain the following further results: The ideal-simpleness, congruence-simpleness, and simpleness of semirings are Morita invariant properties; a complete description of simple semirings containing the infinite element; the "Double Centralizer Property" representation theorem for simple semirings; a complete description of simple semirings containing a projective minimal one-sided ideal; a characterization of ideal-simple semirings having either an infinite element or a projective minimal one-sided ideal; settling a conjecture and a problem as published by Katsov in 2004 for the classes of simple semirings containing either an infinite element or a projective minimal left (right) ideal, showing, respectively, that semirings of those classes are not perfect and that 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, lattice-ordered semirings.
30 citations
Posted Content•
TL;DR: In this article, it was shown that the concepts of simpleness, congruence-simpleness, and idealsimplness for semirings of projective semilattices coincide if and only if the semilatices are finite distributive lattices.
Abstract: 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, congruence-simpleness 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 one-sided ideal; A characterization of ideal-simple semirings having either infinite elements or a projective minimal one-sided ideal; A confirmation of Conjecture of [Kat04a] and solving Problem 3.9 of [Kat04b] 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, lattice-ordered semirings.
19 citations
TL;DR: In this article, the class of semirings S for which the semiirings of square matrices Mn(S) over S are (left) k-artinian is characterized.
Abstract: In this paper we characterize the class of semirings S for which the semirings of square matrices Mn(S) over S are (left) k-artinian. Also an analogue of the Hilbert basis theorem for semirings is obtained.
8 citations
TL;DR: In this paper, it was shown that the left and right operator semiring of a Nobusawa Γ-semiring with unities are Morita equivalent, and the converse was also deduced.
Abstract: We show that the left operator semiring and the right operator semiring of a Nobusawa Γ-semiring with unities are Morita equivalent. The converse is also deduced, i.e., for two Morita equivalent semirings L and R, a Nobusawa Γ-semiring A with unities is constructed such that the left and right operator semirings of A are isomorphic to L and R, respectively. As an application we first establish a relationship between Morita equivalence and Morita context of semirings, and then enumerate some properties of semirings which remain invariant under Morita equivalence.
7 citations