Morita Context Research Articles

Prime structures in a Morita context

In this paper, we study on the primeness and semiprimeness of a Morita context related to the rings and modules. Necessary and sufficient conditions are investigated for an ideal of a Morita context to be a prime ideal and a semiprime ideal. In particular, we determine the conditions under which a Morita context is prime and semiprime.

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.

Multiplicative generalized Lie triple derivations on generalized matrix algebras

Let be a commutative ring with unity, be -algebras, be ()-bimodule and be ()-bimodule. The -algebra is a generalized matrix algebra defined by the Morita context . In this article, we study multiplicative generalized Lie triple derivation on generalized matrix algebras and prove that every multiplicative generalized Lie triple derivation on a generalized matrix algebra has the standard form.

Recollements of abelian categories and ideals in heredity chains—a recursive approach to quasi-hereditary algebras

Recollements of abelian categories are used as a basis of a homological and recursive approach to quasi-hereditary algebras. This yields a homological proof of Dlab and Ringel’s characterisation of idempotent ideals occurring in heredity chains, which in turn characterises quasi-hereditary algebras recursively. Further applications are given to hereditary algebras and to Morita context rings.

Morita invariants of semirings related to a Morita context

The main purpose of the paper is to consider two Morita equivalent semirings [Formula: see text] and [Formula: see text] via Morita context [Formula: see text] instead of considering them via the equivalence of the resulting semimodule categories and then to investigate various Morita invariants related to each of the pairs [Formula: see text]; [Formula: see text]; [Formula: see text]; [Formula: see text], etc.

Rings whose Elements are the Sum of a Tripotent and an Element from the Jacobson Radical

AbstractThis paper is about rings $R$ for which every element is a sum of a tripotent and an element from the Jacobson radical $J(R)$. These rings are called semi-tripotent rings. Examples include Boolean rings, strongly nil-clean rings, strongly 2-nil-clean rings, and semi-boolean rings. Here, many characterizations of semi-tripotent rings are obtained. Necessary and sufficient conditions for a Morita context (respectively, for a group ring of an abelian group or a locally finite nilpotent group) to be semi-tripotent are proved.

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.

Groupoid gradings: A Morita context, semiprimality, and a Bergman-type question

In this paper, we are interested in some natural consequences arising from the canonical duality relation between the concepts of groupoid grading and groupoid action. Any algebra A over a field K graded by a finite groupoid has a canonical structure of a module algebra over the dual algebra of the groupoid algebra . The results here presented concern to the multiplicative structure of the smash product algebra and its representations. An application is the affirmative answer to a Bergmann-type question for groupoid gradings.

Morita context functors on cellular categories

For a linear category over a field K, the morphism space of any two objects admits a bimodule structure over the endomorphism algebras of the objects, so it induces a Morita context between those two algebras. In this article, we use Morita context functors to study cellular categories and give relationships between the cell modules which belong to different endomorphism algebras in a cellular category.

Morita-like equivalence for fair semigroups

In this paper, we mainly investigate Morita-like equivalence and Morita context for right fair semigroups. If two right fair semigroups S and T are Morita-like equivalent, that is, there is a category equivalence F : US − Act : UT-Act : G, we characterize the two functors F and G using Hom functor and tense product functor. Also, we investigate Morita context for right fair semigroups and obtain equivalence between two right unitary act categories.

A Few Remarks on the Tube Algebra of a Monoidal Category

AbstractWe prove two results on the tube algebras of rigid C*-tensor categories. The first is that the tube algebra of the representation category of a compact quantum groupGis a full corner of the Drinfeld double ofG. As an application, we obtain some information on the structure of the tube algebras of the Temperley–Lieb categories 𝒯ℒ(d) ford> 2. The second result is that the tube algebras of weakly Morita equivalent C*-tensor categories are strongly Morita equivalent. The corresponding linking algebra is described as the tube algebra of the 2-category defining the Morita context.

English

We completely determine when a ring consists entirely of weak idempotents, units and nilpotents. We prove that such ring is exactly isomorphic to one of the following: a Boolean ring; Z3 ⊕ Z3; Z3 ⊕ B where B is a Boolean ring; local ring with nil Jacobson radical; M2(Z2) or M2(Z3); or the ring of a Morita context with zero pairings where the underlying rings are Z2 or Z3.

Morita equivalence of semigroups revisited: Firm semigroups

We define firm semigroups and firm acts as non-additive analogues of firm rings and firm modules. Using the categories of firm acts we develop Morita theory for firm semigroups. We show that equivalence functors between categories of firm acts over two firm semigroups have to be tensor multiplication functors. Our main result states that the categories of firm right acts over two firm semigroups are equivalent if and only if these semigroups are strongly Morita equivalent, which means that they are contained in a unitary Morita context with surjective mappings.We also investigate other categories of acts which have been used earlier to develop Morita equivalence. The main tool in our work is adjoint functors. We prove that over firm semigroups all the considered categories are equivalent to the category of firm acts.All this suggests that firm semigroups and firm acts are the natural environment to study Morita equivalence of semigroups.

Gabriel topology related to Morita context of semirings

In this paper we introduce the notion of Gabriel topology on semirings as well as on bisemimodules with unities. We also show that there is a lattice isomorphism between the Gabriel topologies on semirings and those on bisemimodules connected via Morita context.

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.

Quasipolarity of special Morita context rings

Quasipolarity of special Morita context rings

Morita invariants of semirings-II

The purpose of this paper is to investigate the counterparts of most of our results on Morita invariants of semirings for semimodules. Among others, we obtain the lattice isomorphisms between the set of all ideals and set of all subsemimodules corresponding to a Morita context of semirings. Also, the preservation of congruences between a semiring and a semimodule corresponding to a Morita context of semirings is established.

RINGS WITH MANY REGULAR ELEMENTS

In this paper we introduce rings that satisfy regular 1-stable range. These rings are left-right symmetric and are generalizations of unit 1-stable range. We investigate characterizations of these kind of rings and show that these rings are closed under matrix rings and Morita Context rings.

UNIT-REGULAR MODULES

AbstractIn 2014, the first two authors proved an extension to modules of a theorem of Camillo and Yu that an exchange ring has stable range 1 if and only if every regular element is unit-regular. Here, we give a Morita context version of a stronger theorem. The definition of regular elements in a module goes back to Zelmanowitz in 1972, but the notion of a unit-regular element in a module is new. In this paper, we study unit-regular elements and give several characterizations of them in terms of “stable” elements and “lifting” elements. Along the way, we give natural extensions to the module case of many results about unit-regular rings. The paper concludes with a discussion of when the endomorphism ring of a unit-regular module is a unit-regular ring.

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.