Morita Contexts Research Articles

Special Properties of Morita Contexts

In this paper we continue the study of various ring theoretic properties of Morita contexts. Necessary and sufficient conditions are obtained for a general Morita context or a trivial Morita context or a formal triangular matrix ring to satisfy a certain ring property which is among being Kasch, completely primary, quasi-duo, 2-primal, NI, semiprimitive, projective-free, etc. We also characterize when a general Morita context is weakly principally quasi-Baer or strongly right mininjective.

Strongly stratifying ideals, Morita contexts and Hochschild homology

We consider stratifying ideals of finite dimensional algebras in relation with Morita contexts. A Morita context is an algebra built on a data consisting of two algebras, two bimodules and two morphisms. For a strongly stratifying Morita context - or equivalently for a strongly stratifying ideal - we show that Han's conjecture holds if and only if it holds for the diagonal subalgebra. The main tool is the Jacobi-Zariski long exact sequence. One of the main consequences is that Han's conjecture holds for an algebra admitting a strongly (co-)stratifying chain whose steps verify Han's conjecture.If Han's conjecture is true for local algebras and an algebra Λ admits a primitive strongly (co-)stratifying chain, then Han's conjecture holds for Λ.

Enlargements of Quantales

Abstract In this paper, we study the enlargements of quantales. We prove three main results. First, if Q is a factorizable quantale, then any matrix quantale over Q is an enlargement of Q; second, any unital Rees matrix quantale over a quantale Q with an identity is an enlargement of Q; third, two quantales are Morita equivalent if and only if they have a joint enlargement. To prove these theorems, we use quantale matrices and modules and Morita contexts of quantales. Our main theorems and their proofs are parallel to those known for idempotent rings.

When nilpotent elements generate nilpotent ideals

We study the natural class of rings where each nilpotent element generates a nilpotent ideal, calling them the strongly 2-primal rings. We derive many basic properties of these rings, analyze their behavior under standard ring constructions and extensions, and taxonomize their relationship to other natural generalizations of commutativity. A slightly stronger condition is to assume that any nilpotent element generates a nilpotent ideal of the same index of nilpotence. We find that the difference between these two properties explains divergent behaviors in direct products, polynomial rings, Morita contexts, and other constructions.

Some Galois connections arising from Morita contexts of semigroups

We show that a unitary surjective Morita context connecting two semigroups yields Galois connections between certain lattices of compatible relations whenever either semigroup has common weak local units. In the event both semigroups have common weak local units, we obtain mutually inverse lattice isomorphisms that preserve reflexivity, symmetricity and transitivity between the lattices of compatible relations on the semigroups.

A characterization of Goldie extending trivial Morita contexts

A module [Formula: see text] is said to be extending (Goldie extending) if for each submodule [Formula: see text] there exists a direct summand [Formula: see text] of [Formula: see text] such that [Formula: see text] is essential in [Formula: see text] (if for every submodule [Formula: see text] of [Formula: see text] there exists a direct summand [Formula: see text] of [Formula: see text] such that [Formula: see text] is essential in both [Formula: see text] and [Formula: see text]). In this paper, a necessary and sufficient condition is obtained for a trivial Morita context ring to be right Goldie-extending.

Morita contexts and the projective socle property

A ring [Formula: see text] is called a left [Formula: see text]-ring if its left socle, [Formula: see text], is projective. Equivalently, [Formula: see text] is left [Formula: see text]-ring if the right annihilator of every maximal left ideal is of the form [Formula: see text], where [Formula: see text] is an idempotent in [Formula: see text]. In this paper, we characterize when a (trivial) Morita context or a generalized triangular matrix ring is a left [Formula: see text]-ring. Examples to illustrate and delimit the theory are provided.

Rings of Morita contexts which are Dubrovin valuation rings

Let [Formula: see text] be a ring of the Morita context. We define a concept of Dubrovin valuation modules for bi-modules [Formula: see text] and [Formula: see text] and give necessary and sufficient conditions for [Formula: see text] to be a Dubrovin valuation ring in terms of [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text].

On UQ rings

In this paper, we introduce [Formula: see text] rings. A ring [Formula: see text] is called [Formula: see text] if every non-unit element of [Formula: see text] can be represented as a product of a unit and a quasiregular element. We provide various properties of [Formula: see text] rings along with its characterizations. We give a new characterization of [Formula: see text]-good rings, and it turns out that [Formula: see text]-good rings are precisely the rings in which every element is a product of a unit and a quasiregular element. We discuss various extensions of [Formula: see text] rings such as Morita contexts, generalized matrix rings, formal matrix rings, group rings, etc.

Rings whose elements are sums of m-potents and nilpotents

For a fixed natural number an element a of a ring R is m-nil clean if where and b is nilpotent; if further eb = be, a is called strongly m-nil clean. The ring R is called m-nil clean (resp., strongly m-nil clean) if each of its elements is m-nil clean (resp., strongly m-nil clean). For a fixed natural number the strongly m-nil clean rings are a big subclass of the periodic rings which contains the class of strongly nil clean rings. The strongly m-nil cleanness of the tensor product of algebras, matrix rings, Morita contexts, and group rings is discussed in detail. Our results extend several existing results. Examples are provided to illustrate our results.

A characterization of extending trivial Morita contexts

A right module [Formula: see text] is extending if every submodule is essential in a direct summand of [Formula: see text]. In this paper, necessary and sufficient conditions are obtained for a trivial Morita context ring to be right extending.

Reduced rings and modules arising from Morita contexts

Reduced rings and modules arising from Morita contexts

Morita contexts and unitary ideals of rings

Morita contexts and unitary ideals of rings

Jordan Automorphism of Morita Context Algebras

The aim of this article is to determine entirely the Jordan automorphisms of generalized matrix rings of Morita contexts. Necessary and sufficient conditions are obtained for an $$\mathcal {R}$$ -linear map on a general Morita context to be a Jordan homomorphism. Moreover, some conditions are studied, under which, any Jordan automorphism of a general Morita context is either an automorphism or an anti-automorphism.

Frobenius cowreaths and Morita contexts

Abstract 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.

Lattices and quantales of ideals of semigroups and their preservation under Morita contexts

We study properties of the lattice of unitary ideals of a semigroup. In particular, we show that it is a quantale. We prove that if two semigroups are connected by an acceptable Morita context then there is an isomorphism between the quantales of unitary ideals of these semigroups. Moreover, factorisable ideals corresponding to each other under this isomorphism are strongly Morita equivalent.

A Characterization of G-Dedekind Prime Morita Contexts

Let $T=\left (\begin {array}{cc} R & V\\ W & S \end {array}\right )$ be the ring of a Morita context such that VW = R and WV = S. In this paper, we give necessary and sufficient conditions for T to be a G-Dedekind prime ring.

On 2-nil-good rings

A ring [Formula: see text] is defined to be 2-nil-good if every element in [Formula: see text] is the sum of two units and a nilpotent. Fundamental properties of such rings are obtained. We prove that every strongly [Formula: see text]-regular ring is 2-nil-good if and only if the identity is the sum of two units. One of the main result of this paper is that every square matrix ring over J-fine rings is 2-nil-good. We establish 2-nil-good property for Morita contexts. This implies, in particular, that every matrix ring over 2-nil-good rings is 2-nil-good. Furthermore, we prove that the ring [Formula: see text] of all lower triangular diagonal-finite matrices over a 2-good ring [Formula: see text] is 2-nil-good.

The Morita contexts and galois extensions for weak Hopf group coalgebras

In this paper, we mainly construct a Morita context associated to a weak Hopf group coalgebra and as application we also compute the Morita contexts of weak Hopf group Galois extensions.

Morita contexts, ideals, and congruences for semirings with local units

Morita contexts, ideals, and congruences for semirings with local units