Morita Context Research Articles

Generalised Lat-Igusa-Todorov Algebras and Morita Contexts

AbstractIn this paper we define (special) GLIT classes and (special) GLIT algebras. We prove that GLIT algebras, which generalise Lat-Igusa-Todorov algebras, satisfy the finitistic dimension conjecture and give several properties and examples. In addition we show that special GLIT algebras are exactly those that have finite finitistic dimension. Lastly we study Morita algebras arising from a Morita context and give conditions for them to be (special) GLIT in terms of the algebras and bimodules used in their definition. As a consequence we obtain simple conditions for a triangular matrix algebra to be (special) GLIT and also prove that the tensor product of a GLIT $$\mathbb {K}$$ K -algebra with a path algebra of a finite quiver without oriented cycles is GLIT.

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.

Quasi-hereditary rings are closed under taking block extensions

In this paper, we give a sufficient condition for Morita context rings to be quasi-hereditary. As an application, we show that each block extension of a quasi-hereditary ring is also quasi-hereditary.

Lie centralizing mappings on generalized matrix algebras through two-sided zero products

Let [Formula: see text] be a generalized matrix algebra defined by the Morita context [Formula: see text] and [Formula: see text] the center of [Formula: see text]. In this paper, under some certain conditions on [Formula: see text], we prove that if [Formula: see text] is an additive map satisfying [Formula: see text] for any [Formula: see text] with [Formula: see text], then [Formula: see text] has the form [Formula: see text] for all [Formula: see text], where [Formula: see text] and [Formula: see text] is an additive map from [Formula: see text] into [Formula: see text]. Finally as its applications, we characterize Lie centralizer maps and generalized Lie derivations on [Formula: see text]. Moreover we prove that the similar conclusions remain valid on full matrix algebras, triangular algebras, upper triangular matrix algebras.

Torsion pairs and related modules over trivial ring extensions

Let $R\ltimes M$ be a trivial extension of a ring $R$ by an $R$-$R$-bimodule $M$. We first study how to construct torsion pairs over $R\ltimes M$ from torsion pairs over $R$. Some characterizations of finitely generated (presented) modules, flat modules and coherent rings relative to a torsion pair over $R\ltimes M$ are obtained. Then we discuss the transfers of torsion pairs over $R\ltimes M$ to $R$. Finally, some applications are given in Morita context rings.

Lie (Jordan) $ \sigma- $centralizer at the zero products on generalized matrix algebra

<p>Given a unital commutative ring $ \mathscr{R} $, $ (\mathscr{A}, \mathscr{B}) $ and $ (\mathscr{B}, \mathscr{A}) $ are bimodules of $ \mathscr{M} $ and $ \mathscr{N} $, respectively, where $ \mathscr{A}, \mathscr{B} $ are unitals $ \mathscr{R}- $algebras. The $ \mathscr{R}- $algebra $ \mathscr{G} = $ $ \mathscr{G}(\mathscr{A}, \mathscr{M}, \mathscr{N}, \mathscr{B}) $ is a generalized matrix algebra described by the Morita context $ (\mathscr{A}, \mathscr{B}, \mathscr{M}, \mathscr{N}, \zeta_{\mathscr{M}\mathscr{N}}, \chi_{\mathscr{N}\mathscr{M}}) $. The present study investigated the structure of Lie (Jordan) $ \sigma- $centralizers at the zero products on order two generalized matrix algebra and established that each Jordan $ \sigma- $centralizer at the zero products is a $ \sigma- $centralizer at the zero product on order two generalized matrix algebra. We also provided sufficient and necessary conditions under which a Lie $ \sigma- $centralizer at the zero product is proper on an order two generalized matrix algebra.</p>

Gorenstein projective modules over Milnor squares of rings

<p>We construct a class of Gorenstein-projective modules over Milnor squares of rings. As an application, we obtain Gorenstein-projective modules over Morita context rings with two bimodule homomorphisms zero in the general setting instead of Artin algebras or Noetherian rings.</p>

2-Primal semimodule related to a Morita context

In this paper we introduce the notion of 2-primal semimodule related to a Morita context < R, S,R PS,S QR, θ, ϕ >. We obtain some characterizations of 2-primal semimodule.

On connections between Morita semigroups and strong Morita equivalence

A surjective Morita context connecting semigroups S and T yields a Morita semigroup and a strict local isomorphism from it onto S along which idempotents lift. We describe strong Morita equivalence of firm semigroups in terms of Morita semigroups and isomorphisms. We also generalize some of Hotzel’s theorems to semigroups with weak local units. In particular, the Morita semigroup induced by a dual pair β over a semigroup with weak local units is isomorphic to Σ β .

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.

Cotorsion pairs and Hovey triples over trivial ring extensions

Let be a trivial extension of a ring R by an R-R-bimodule M. We first give some homological formulas over . Then we investigate how to construct (hereditary, complete) cotorsion pairs and (hereditary, cofibrantly generated) Hovey triples over . Finally, some applications are given in some Morita context rings.

Ding projective modules over Morita context rings

– Let be a Morita context ring with zero bimodule homomorphisms. We give sufficient conditions for modules over to be Ding projective.

On BiHom-L-R Smash Products

Let [Formula: see text] be a BiHom-Hopf algebra and [Formula: see text] be an [Formula: see text]-BiHom-bimodule algebra, where the maps [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] are bijective. We first prove the Maschke-type theorem for the BiHom-L-R smash product over a finite-dimensional semisimple BiHom-Hopf algebra. Next we give a Morita context between the BiHom-subalgebra [Formula: see text] and the BiHom-L-R smash product [Formula: see text].

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.

A Morita-Takeuchi Context and Hopf Coquasigroup Galois Coextensions

For H a Hopf quasigroup and C, a left quasi H-comodule coalgebra, we show that the smash coproduct C⋊H (as a symmetry of smash product) is linked to some quotient coalgebra Q=C/CH*+ by a Morita-Takeuchi context (as a symmetry of Morita context). We use the Morita-Takeuchi setting to prove that for finite dimensional H, equivalent conditions for C/Q to be a Hopf quasigroup Galois coextension (as a symmetry of Galois extension). In particular, we consider a special case of quasigroup graded coalgebras as an application of our theory.

Gorenstein flat modules over some Morita context rings

Let [Formula: see text] be a Morita context ring with zero bimodule homomorphisms. We establish sufficient and necessary conditions for modules over [Formula: see text] to be Gorenstein flat.

Hopf Quasigroup Galois Extensions and a Morita Equivalence

For H, a Hopf coquasigroup, and A, a left quasi-H-module algebra, we show that the smash product A#H is linked to the algebra of H invariants AH by a Morita context. We use the Morita setting to prove that for finite dimensional H, there are equivalent conditions for A/AH to be Galois parallel in the case of H finite dimensional Hopf algebra.

A study on generalized matrix algebras having generalized Lie derivations

Let R be a commutative ring with unity. The R-algebra G=G(A,M,N,B) is a generalized matrix algebra defined by the Morita context (A,B,M,N, ξMN,ΩNM). In this article, westudy generalized Lie derivation and show that every generalized Lie derivation on a generalized matrix algebra has the standard form under certain assumptions.

On regularity and intra regularity of the components of a Morita context of monoids

The notions of regular, intra regular and duo bi-acts of a Morita context of monoids have been introduced. Some characterizations of regular and intra regular bi-acts have been obtained in terms of b-subacts. Also connections among regularity of acts and [Formula: see text]-regularity of power semimodules, intra regularity of acts and intra [Formula: see text]-regularity of power semimodules have been observed using power functor. Then the study of the interplay among various components of a Morita context of monoids in terms of regularity, intra regularity and duoness have been accomplished. It has been deduced that regularity, intra regularity and being duo are Morita invariant properties. Finally regular (intra regular, duo) Morita context has been defined and it has been proved that for a duo Morita context of monoids regular and intra regular Morita context coincide.