Congruence-Simple Semirings

Bashir, Robert El; Kepka, Tomáš

doi:10.1007/s00233-007-0725-7

Cited by 27 publications

(20 citation statements)

References 9 publications

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“…Following [5], a semiring S is congruence-simple if the diagonal, △ S , and the universal, S 2 , congruences are the only congruences on S; and S is ideal-simple if 0 and S are its only ideals. A semiring S is said to be simple if it is simultaneously congruence-simple and ideal-simple.…”

Section: Preliminariesmentioning

confidence: 99%

“…Also, 'congruence-and ideal-simple semirings' constitutes another booming area in semiring research which has quite interesting and promising applications in various fields, in particular in cryptography [38] (for some relatively recent developments in this area we refer our potential readers to [4], [5], [21], [22], [23], [30], [31], [32], [39], [40], and [45]). In this respect, in the present paper we consider, in the context of CP-semirings, congruenceand ideal-simple semirings as well.…”

Section: Introductionmentioning

confidence: 99%

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Toward homological structure theory of semimodules: On semirings all of whose cyclic semimodules are projective

2017

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In this paper, we introduce homological structure theory of semirings and CP-semirings-semirings all of whose cyclic semimodules are projective. We completely describe semisimple, Gelfand, subtractive, and anti-bounded, CP-semirings. We give complete characterizations of congruence-simple subtractive and congruence-simple anti-bounded CP-semirings, which solve two earlier open problems for these classes of semirings. We also study in detail the properties of semimodules over Boolean algebras whose endomorphism semirings are CP-semirings; and, as a consequence of this result, we give a complete description of ideal-simple CP-semirings.

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Section: Preliminariesmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Toward homological structure theory of semimodules: On semirings all of whose cyclic semimodules are projective

2017

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“…A hemiring R is congruence-simple [1] (also, [2], [24]) if the diagonal, △ R , and universal, R 2 , congruences are the only congruences on R. In the next theorem, we, in particular, give a complete description of J-semisimple, congruence-simple hemirings.…”

Section: The Jacobson Type Radicals Of Semiringsmentioning

confidence: 99%

On Radicals of Semirings and Related Problems

Katsov

Nam

2014

Communications in Algebra

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The aim of this paper is to develop an 'external' Kurosh-Amitsur radical theory of semirings and, using this approach, to obtain some fundamental results regarding two Jacobson type of radicals -the Jacobson-Bourne, J-, radical and a very natural its variation, J sradical -of hemirings, as well as the Brown-McCoy, R BM -, radical of hemirings. Among the new central results of the paper, we single out the following ones: Theorems unifying two, internal and external, approches to the Kurosh-Amitzur radical theory of hemirings; A characterization of J-semisimple hemirings; A description of J-semisimple congruence-simple hemirings; A characterization of finite additively-idempotent J s -semisimple hemirings; Complete discriptions of R BM -semisimple commutative and latticeordered hemirings; Semiring versions of the well-known classical ring results-Nakayama's and Hopkins Lemmas and Jacobson-Chevalley Density Theorem; Establishing the fundamental relationship between the radicals J, J s , and R BM of hemirings R and matrix hemirings M n (R); Establishing the matric-extensibleness (see, e.g., [4, Section 4.9]) of the radical classes of the Jacobson, Brown-McCoy, and J s -, radicals of hemirings; Showing that the J-semisimplicity, J s -semisimplicity, and R BM -semisimplicity of semirings are Morita invariant properties.

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“…Investigating semirings and their representations, one should undoubtedly use methods and techniques of both ring and lattice theory as well as diverse techniques and methods of categorical and universal algebra. Thus, the wide variety of the algebraic techniques involved in studying semirings, and their representations/semimodules, perhaps explains why the research on simple semirings is still behind of that for rings and groups (for some recent activity and results on this subject one may consult [26], [1], [27], [2], [14], [31], [13], and [20]).…”

Section: Introductionmentioning

confidence: 99%

On simpleness of semirings and complete semirings

2014

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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.

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Congruence-Simple Semirings

Cited by 27 publications

References 9 publications

Toward homological structure theory of semimodules: On semirings all of whose cyclic semimodules are projective

Toward homological structure theory of semimodules: On semirings all of whose cyclic semimodules are projective

On Radicals of Semirings and Related Problems

On simpleness of semirings and complete semirings

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