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Morita equivalence and homological characterization of semirings. (English) Zbl 1250.16031

For any additively commutative semiring \((S,+,\cdot)\) with absorbing zero and identity, \(_S\mathcal M\) (resp. \(\mathcal M_S\)) denotes the category of left (resp. right) \(S\)-semimodules. Semirings \(S\) and \(T\) as above are called Morita equivalent if there exists a finitely generated projective generator \(_SP\) for \(_S\mathcal M\) such that the semirings \(T\) and \(\text{End}(_SP)\) are isomorphic. Then it is shown that this is indeed an equivalence relation on the category of all semirings, and that semirings \(S\) and \(T\) are Morita equivalent if and only if the categories \(_S\mathcal M\) and \(_T\mathcal M\) (resp. \(\mathcal M_S\) and \(\mathcal M_T\)) are equivalent. From this result several characterizations of semisimple rings \(R\) among all semisimple semirings (resp. among all right (left) subtractive semirings) by properties of their \(R\)-semimodules are obtained. This leads to characterizations of division rings \(D\) in the class of all division semirings in a similar way. Finally, some additional results are proved in the case that \(R\) is an additively regular semisimple semiring (resp. \(D\) is an additively regular division semiring).

MSC:

16Y60 Semirings
16D90 Module categories in associative algebras
18A40 Adjoint functors (universal constructions, reflective subcategories, Kan extensions, etc.)
18G05 Projectives and injectives (category-theoretic aspects)
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