Morita Contexts Research Articles

INVERTIBLE BIMODULES, MIYASHITA ACTION IN MONOIDAL CATEGORIES AND AZUMAYA MONOIDS

In this paper we introduce and study Miyashita action in the context of monoidal categories aiming by this to provide a common framework of previous studies in the literature. We make a special emphasis of this action on Azumaya monoids. To this end, we develop the theory of invertible bimodules over different monoids (a sort of Morita contexts) in general monoidal categories as well as their corresponding Miyashita action. Roughly speaking, a Miyashita action is a homomorphism of groups from the group of all isomorphic classes of invertible subobjects of a given monoid to its group of automorphisms. In the symmetric case, we show that for certain Azumaya monoids, which are abundant in practice, the corresponding Miyashita action is always an isomorphism of groups. This generalizes Miyashita’s classical result and sheds light on other applications of geometric nature which cannot be treated using the classical theory. In order to illustrate our methods, we give a concrete application to the category of comodules over commutative (flat) Hopf algebroids. This obviously includes the special cases of split Hopf algebroids (action groupoids), which for instance cover the situation of the action of an affine algebraic group on an affine algebraic variety.

Rings of Morita contexts which are maximal orders

Let [Formula: see text] be a ring of Morita context which is a prime Goldie ring with its quotient ring [Formula: see text]. We define the notion of an [Formula: see text]-maximal module in [Formula: see text] and that of an [Formula: see text]-maximal module in [Formula: see text] from order theoretical point of view and give some necessary and sufficient conditions for [Formula: see text] to be a maximal order in terms of [Formula: see text]-module [Formula: see text] and [Formula: see text]-module [Formula: see text]. In case [Formula: see text] is a maximal order, we explicitly describe the structure of [Formula: see text]-[Formula: see text]-ideals. These results are applied to obtain necessary and sufficient conditions for [Formula: see text] to be an Asano order or a Dedekind order.

Wide right Morita contexts in lax-unital bicategories

The classical result in the theory of unital rings that the maps of a Morita context are isomorphisms when they are epimorphisms can be proven in the general setting of wide right Morita contexts in bicategories. There exists a similar result for non-unital rings, but bicategories are not general enough to handle that case. In this paper, we use the more general lax-unital bicategories to prove a version of the result and study some related questions.

Nil-clean and strongly nil-clean rings

An element a of a ring R is nil-clean if a=e+b where e2=e∈R and b is a nilpotent; if further eb=be, the element a is called strongly nil-clean. The ring R is called nil-clean (resp., strongly nil-clean) if each of its elements is nil-clean (resp., strongly nil-clean). It is proved that an element a is strongly nil-clean iff a is a sum of an idempotent and a unit that commute and a−a2 is a nilpotent, and that a ring R is strongly nil-clean iff R/J(R) is boolean and J(R) is nil, where J(R) denotes the Jacobson radical of R. The strong nil-cleanness of Morita contexts, formal matrix rings and group rings is discussed in details. A necessary and sufficient condition is obtained for an ideal I of R to have the property that R/I strongly nil-clean implies R is strongly nil-clean. Finally, responding to the question of when a matrix ring is nil-clean, we prove that the matrix ring over a 2-primal ring R is nil-clean iff R/J(R) is boolean and J(R) is nil, i.e., R is strongly nil-clean.

Morita Contexts and Partial Group Galois Extensions for Hopf Group Coalgebras

We first introduce the notion of partial group twisted smash product and construct a Morita context for partial coaction of co-Frobenius Hopf group coalgebra relating generalized partial group smash product and partial coinvariants. Furthermore, we study group-Galois extensions and reobtain some classical results in the partial case. Finally, we prove that any partial group Galois extension induces a unique partial group entwining map compatible with the partial right coaction.

Morita Theory for Group Corings

Using the theory of group corings, we study (graded) Morita contexts associated to a comodule over a group coring, which generalize and unify some classical morita contexts. Some applications of our theory are also discussed.

Study of Morita contexts

This article concerns mainly on various ring properties of Morita contexts. Necessary and sufficient conditions are obtained for a general Morita context or a trivial Morita context or a generalized matrix ring over a ring to satisfy a certain ring property which is among being semilocal, semiperfect, left perfect, semiprimary, semipotent, potent, clean, strongly π-regular, semiregular, etc. Many known results on a formal triangular matrix ring are extended to a Morita context or a trivial Morita context. Some questions on this subject raised by Varadarajan in [22] are answered.

On Artin Algebras Arising from Morita Contexts

We study Morita rings \(\Lambda _{(\phi ,\psi )}=\left (\begin {array}{cc}A &_{A}N_{B} _{B}M_{A} & B \end {array}\right )\) in the context of Artin algebras from various perspectives. First we study covariantly finite, contravariantly finite, and functorially finite subcategories of the module category of a Morita ring when the bimodule homomorphisms \(\phi \) and \(\psi \) are zero. Further we give bounds for the global dimension of a Morita ring \(\Lambda _{(0,0)}\), as an Artin algebra, in terms of the global dimensions of A and B in the case when both \(\phi \) and \(\psi \) are zero. We illustrate our bounds with some examples. Finally we investigate when a Morita ring is a Gorenstein Artin algebra and then we determine all the Gorenstein-projective modules over the Morita ring \(\Lambda _{\phi ,\psi }\) in case \(A=N=M=B\) and A an Artin algebra.

Morita Contexts as Lax Functors

Monads are well known to be equivalent to lax functors out of the terminal category. Morita contexts are here shown to be lax functors out of the chaotic category with two objects. This allows various aspects in the theory of Morita contexts to be seen as special cases of general results about lax functors. The account we give of this could serve as an introduction to lax functors for those familiar with the theory of monads. We also prove some very general results along these lines relative to a given 2-comonad, with the classical case of ordinary monad theory amounting to the case of the identity comonad on Cat.

On Almost Unit-Regular Rings

An element a ∈ R is unit-regular provided that there exists an invertible u ∈ R such that a = aua. A ring R is called an almost unit-regular ring provided that for any a ∈ R, either a or 1 − a is unit-regular. We characterize, in this article, the almost unit-regularity of Morita contexts with zero pairings. We also show that a ring R is unit-regular if and only if M 2(R) is almost unit-regular. Various examples of such rings are constructed by means of formal triangular matrix rings.

Generalized Stable Rings and Regularity

We prove in this article that the generalized stable property is invariant under Morita contexts. Further, we show that many classes of square matrices over generalized stable regular rings admit a diagonal reduction. Related examples are constructed as well.

On Morita Contexts in Bicategories

We characterize abstract Morita contexts in several bicategories. In particular, we use heteromorphisms for the bicategory \(\mathbb{Prof} \) of categories and profunctors and coreflective subcategories for \(\mathbb {Cat}\) (categories and functors). In addition, we prove general statements concerning strict Morita contexts, and we give new equivalent forms to the standard notions of adjointness, category equivalence and Morita equivalence by studying the collage of a profunctor.

Strong Morita equivalence of semigroups with local units

In this paper we study Morita contexts for semigroups. We prove a Rees matrix cover connection between strongly Morita equivalent semigroups and investigate how the existence of a unitary Morita semigroup over a given semigroup is related to the existence of a ‘good’ Rees matrix cover of this semigroup.

Rings of the Morita Contexts and Semihereditary Maximal V-Orders

Some necessary and sufficient conditions are given for rings of the Morita contexts to be semihereditary maximal V-orders, Prüfer rings and Dubrovin valuation rings.

The Eilenberg-Moore Category and a Beck-type Theorem for a Morita Context

The Eilenberg-Moore constructions and a Beck-type theorem for pairs of monads are described. More specifically, a notion of a Morita context comprising of two monads, two bialgebra functors and two connecting maps is introduced. It is shown that in many cases equivalences between categories of algebras are induced by such Morita contexts. The Eilenberg-Moore category of representations of a Morita context is constructed. This construction allows one to associate two pairs of adjoint functors with right adjoint functors having a common domain or a double adjunction to a Morita context. It is shown that, conversely, every Morita context arises from a double adjunction. The comparison functor between the domain of right adjoint functors in a double adjunction and the Eilenberg-Moore category of the associated Morita context is defined. The sufficient and necessary conditions for this comparison functor to be an equivalence (or for the moritability of a pair of functors with a common domain) are derived.

Morita Theory for Comodules Over Corings

By a theorem due to Kato and Ohtake, any (not necessarily strict) Morita context induces an equivalence between appropriate subcategories of the module categories of the two rings in the Morita context. These are in fact categories of firm modules for non-unital subrings. We apply this result to various Morita contexts associated to a comodule Σ of an A-coring 𝒞. This allows to extend (weak and strong) structure theorems in the literature, in particular beyond the cases when any of the coring 𝒞 or the comodule Σ is finitely generated and projective as an A-module. That is, we obtain relations between the category of 𝒞-comodules and the category of firm modules for a firm ring R, which is an ideal of the endomorphism algebra End 𝒞(Σ). For a firmly projective comodule of a coseparable coring we prove a strong structure theorem assuming only surjectivity of the canonical map.

Co-Frobenius corings and adjoint functors

We study co-Frobenius and more generally quasi-co-Frobenius corings over arbitrary base rings and over PF base rings in particular. We generalize some results about co-Frobenius and quasi-co-Frobenius coalgebras to the case of non-commutative base rings and give several new characterizations for co-Frobenius and more generally quasi-co-Frobenius corings, some of them are new even in the coalgebra situation. We construct Morita contexts to study Frobenius properties of corings and a second kind of Morita contexts to study adjoint pairs. Comparing both Morita contexts, we obtain our main result that characterizes quasi-co-Frobenius corings in terms of a pair of adjoint functors ( F , G ) such that ( G , F ) is locally quasi-adjoint in a sense defined in this note.

Morita Contexts and their Lattices of Relations

We study the interaction between the lattices of relations of members of a general Morita context. The pairs of reversing-order maps are defined, which determine the dualities between the lattices of ‘closed’ relations. Under rather weak conditions, these dualities can be composed obtaining the projectivities defined by simple maps.

Morita contexts on the Hopf algebroid

Morita contexts on the Hopf algebroid

Extensions of GM-rings

It is shown that a ring R is a GM-ring if and only if there exists a complete orthogonal set { e 1,...,e n } of idempotents such that all e i Re i are GM-rings. We also investigate GM-rings for Morita contexts, module extensions and power series rings.