Injective and projective acts over a completely 0-simple semigroup

I. B. Kozhukhov ; A. O. Petrikov

Čebyševskij sbornik, Tome 17 (2016) no. 4, pp. 65-78.

Résumé

The homological theory of rings and modules is an important branch of algebra. It provided answers to numerous questions of the theory of rings. Along with the homological theory, another theory started to develop, also under significant influence of the theory of rings, which is the homological theory of universal algebras, and, in particular, of semigroups and acts over them. This theory analyses such notions as injective and projective acts over semigroups, injective hulls and projective covers. As in the case of rings and modules, the injective hull exists for every act, while the projective cover sometimes does not. In 1967 P. Berthiaume proved the existence of injective hulls of an arbitrary act over a semigroup (without the assumption of the presence of an identity in the semigroup). J. Isbell studied monoids (i.e. semigroups with an identity) over which every act has a projective cover. L. A. Skornyakov developed a homological theory of monoids. Many results of that theory were mentioned in the known monograph by M. Kilp, U. Knauer, A. V. Mikhalev. For semigroups of a relatively simple structure the results of the homological theory can be significantly refined. For example, in 2012 G. Moghaddasi described injective acts and built injective hulls of acts over a left zero semigroup assuming the separability of the act. I. B. Kozhukhov and A. P. Haliullina described injective and projective acts over groups and right zero semigroups, built injective hulls and projective covers of acts over such semigroups. For acts over a left zero semigroup the condition of separability of acts was removed. An important class of semigroups containing groups, left and right zero semigroups, rectangular bands is the class of completely simple semigroups, as well as the broader class of completely $0$-simple semigroups. In 2000 A. Yu. Avdeyev and I. B. Kozhukhov described all acts over completely simple semigroups and acts with zero over completely $0$-simple semigroups. It triggered further reasearch of acts over such semigroups. I. B. Kozhuhov and A. O. Petrikov described injective and projective acts over completely simple semigroups, thereby generalising the results of I. B. Kozhuhov and A. R. Khaliullina, and also the work of G. Mogaddasi. They built injective hulls and projective covers of acts over such semigroups. In this paper the above-mentioned results concerning acts over completely simple semigroups were generalized to acts with zero over completely $0$-simple semigroups. In particular, the necessary and sufficient conditions of injectivity and projectivity of an act with zero over an arbitrary completely $0$-simple semigroup were found, injective hulls and projective covers of arbitrary acts with zero over such semigroups were built. It was established that a projective act over an arbitrary completely $0$-simple semigroup is exactly a $0$-coproduct of a free act and acts isomorphic to a $0$-minimal right ideal of the semigroup (considered as a right act). Bibliography: 15 titles.

Détail

Keywords: Act over semigroup, injective act, projective act, completely 0-simple semigroups, injective hull, projective cover.

Bibliographie

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