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We characterize those semigroups with local units which are Morita equivalent to semigroups with local units having commuting idempotents. A key element in this characterization is a description of those inverse categories which are equivalent to Cauchy completions of inverse semigroups. [ABSTRACT FROM PUBLISHER]

We introduce an equivalence between central simple strongly G-graded algebras. Such classes are associated in a natural way to absolutely irreducible characters of semisimple G-graded algebras. We study invariants of this equivalence relation, and also the structure of certain representatives of the equivalence classes. [ABSTRACT FROM AUTHOR]

Abstract: A Morita context is constructed for any comodule of a coring and, more generally, for an bicomodule Σ for a coring extension of . It is related to a 2-object subcategory of the category of k-linear functors . Strictness of the Morita context is shown to imply the Galois property of Σ as a -comodule and a Weak Structure Theorem. Sufficient conditions are found also for a Strong Structure Theorem to hold. Cleft property of an bicomodule Σ—implying strictness of the associated Morita context—is introduced. It is shown to be equivalent to being a Galois -comodule and isomorphic to , in the category of left modules for the ring and right comodules for the coring , i.e. satisfying the normal basis property. Algebra extensions, that are cleft extensions by a Hopf algebra, a coalgebra or a Hopf algebroid, as well as cleft entwining structures (over commutative or non-commutative base rings) and cleft weak entwining structures, are shown to provide examples of cleft bicomodules. [Copyright &y& Elsevier]

Abstract: We apply ideas from commutative algebra, and Morita theory to algebraic topology using ring spectra. This allows us to prove new duality results in algebra and topology, and to view (1) Poincaré duality for manifolds, (2) Gorenstein duality for commutative rings, (3) Benson–Carlson duality for cohomology rings of finite groups, (4) Poincaré duality for groups and (5) Gross–Hopkins duality in chromatic stable homotopy theory as examples of a single phenomenon. [Copyright &y& Elsevier]