The fineness properties of Morita contexts

Let [Formula: see text] be a Morita context. For generalized fine (respectively, generalized unit-fine) rings [Formula: see text] and [Formula: see text], it is proved that [Formula: see text] is generalized fine (respectively, generalized unit-fine) if and only if, for [Formula: see text] and [Formula: see text], [Formula: see text] implies [Formula: see text] and [Formula: see text] implies [Formula: see text]. Especially, for fine (respectively, unit-fine) rings [Formula: see text] and [Formula: see text], [Formula: see text] is fine (respectively, unit-fine) if and only if, for [Formula: see text] and [Formula: see text], [Formula: see text] implies [Formula: see text] and [Formula: see text] implies [Formula: see text]. As consequences, (1) matrix rings over fine (respectively, unit-fine, generalized fine and generalized unit-fine) rings are fine (respectively, unit-fine, generalized fine and generalized unit-fine); (2) a sufficient condition for a simple ring to be fine (respectively, unit-fine) is obtained: a simple ring [Formula: see text] is fine (respectively, unit-fine) if both [Formula: see text] and [Formula: see text] are fine (respectively, unit-fine) for some [Formula: see text]; and (3) a question of Cǎlugǎreanu [1] on unit-fine matrix rings is affirmatively answered.

The Maschke-Type Theorem and Morita Context for BiHom-Smash Products

Let H , α H , β H , ω H , ψ H , S H be a BiHom-Hopf algebra and A , α A , β A be an H , α H , β H -module BiHom-algebra. Then, in this paper, we study some properties on the BiHom-smash product A # H . We construct the Maschke-type theorem for the BiHom-smash product A # H and form an associated Morita context A H , A H A A # H , A # H A A H , A # H .

Clean coalgebras and clean comodules of finitely generated projective modules

Let R be a commutative ring with multiplicative identity and P is a finitely generated projective R-module. If P∗ is the set of R-module homomorphism from P to R, then the tensor product P∗⊗RP can be considered as an R-coalgebra. Furthermore, P and P∗ is a comodule over coalgebra P∗⊗RP. Using the Morita context, this paper give sufficient conditions of clean coalgebra P∗⊗RP and clean P∗⊗RP-comodule P and P∗. These sufficient conditions are determined by the conditions of module P and ring R.

On Morita equivalence of partially ordered semigroups with local units

We show that for two partially ordered semigroups S and T with common local units, there exists a unitary Morita context with surjective maps if and only if the categories of closed right S- and T-posets are equivalent.

Frobenius cowreaths and Morita contexts

We prove a uniqueness type theorem for (weak, total) integrals on a Frobenius cowreath in a monoidal category. When the cowreath is, moreover, pre-Galois, we construct a Morita context relating the subalgebra of coinvariants and a certain wreath algebra. Then we see that the strictness of the Morita context is related to the Galois property of the cowreath and the existence of a weak total integral on it. We apply our results to quasi-Hopf algebras.

Partial (Co)actions of multiplier Hopf algebras: Morita and Galois theories

In this work, we deal with partial (co)actions of multiplier Hopf algebras on not necessarily unital algebras. Our main goal is to construct a Morita context relating the coinvariant algebra [Formula: see text] with a certain subalgebra of the smash product [Formula: see text]. Besides that, we present the notion of partial Galois coaction, which is closely related to this Morita context.