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Simple semirings with zero. (English) Zbl 1348.16033

Let \((S,+,\cdot)\) be an additively commutative and idempotent semiring with the natural partial order \(x\leq y\Longleftrightarrow x+y=y\) for all \(x,y\in S\). A left ideal \(K\) of \(S\) is called complete if \(a\in K\) and \(b\leq a\) imply \(b\in K\). A left semimodule \(_SM\), clearly also assumed to be commutative and idempotent, is called faithful if for \(a\neq b\) in \(S\) there is some \(x\in M\) such that \(ax\neq bx\). A faithful left semimodule \(_SM\) is called characteristic if \((M,+)\) has a neutral element \(0\in M\) satisfying \(S0=\{0\}\), and there is a mapping \(\varepsilon\colon M\times M\to S\) such that \(\varepsilon(u,v)x=0\) and \(\varepsilon(u,v)y=v\) for all \(u,v,x,y\in M\), \(x\leq u\), \(y\nleq u\).
Now, let \((S,+,\cdot)\), \(|S|\geq 3\), have an absorbing zero as well as a greatest element. Assume further that for every complete left ideal \(K\) of \(S\) such that \(K\) and \(S\setminus K\) are infinite and there is a greatest element in \(K\). Then the following conditions are equivalent. (i) \(S\) is simple and has at least one minimal left ideal. (ii) \(S\) is simple and there is a faithful minimal semimodule \(_SN\). (iii) There exists some characteristic semimodule \(_SM\).

MSC:

16Y60 Semirings
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[1] 1. R. El Bashir, J. Hurt, A. Jančařík and T. Kepka, Simple commutative semirings, J. Algebra263 (2001) 277-306. genRefLink(16, ’S021949881650047XBIB001’, ’10.1006
[2] 2. R. El Bashir and T. Kepka, Congruence-simple semirings, Semigroup Forum75 (2007) 588-608. genRefLink(16, ’S021949881650047XBIB002’, ’10.1007
[3] 3. J. Golan, The Theory of Semirings with Applications in Mathematics and Theoretical Computer Science, Pitman Monographs, Vol. 54 (Longman, Harlow, 1992). · Zbl 0780.16036
[4] 4. V. Hebisch and H. J. Weinert, Halbringe – Algebraische Theorie und Anwendungen in der Informatik (Teubner, Stuttgart, 1993). · Zbl 0829.16035
[5] 5. A. Kendziorra and J. Zumbrägel, Finite simple additively idempotent semirings, J. Algebra388 (2013) 43-64. genRefLink(16, ’S021949881650047XBIB005’, ’10.1016
[6] 6. G. Maze, C. Monico and J. Rosenthal, Public key cryptography based on semigroup actions, Adv. Math. Commun.1 (2007) 489-507. genRefLink(16, ’S021949881650047XBIB006’, ’10.3934 · Zbl 1194.94190
[7] 7. S. S. Mitchell and P. B. Fenoglio, Congruence-free commutative semirings, Semigroup Forum37 (1988) 79-91. genRefLink(16, ’S021949881650047XBIB007’, ’10.1007
[8] 8. C. Monico, On finite congruence-simple semirings, J. Algebra271 (2004) 846-854. genRefLink(16, ’S021949881650047XBIB008’, ’10.1016
[9] 9. J. Zumbrägel, Classification of finite congruence-simple semirings with zero, J. Algebra Appl.7 (2008) 363-377. [Abstract] genRefLink(128, ’S021949881650047XBIB009’, ’000257114400006’);
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