Morita Context Research Articles

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.

On Twisted Smash Products of Monoidal Hom–Hopf Algebras

Let (H, α) be a monoidal Hom-Hopf algebra and (A, β) be an (H, α)-Hom-bimodule algebra. In this article, we first introduce the notion of a twisted Hom-smash product A☆H and then find some sufficient and necessary conditions making it into a monoidal Hom-bialgebra. Furthermore, we give a Maschke-type theorem for the twisted Hom-smash product over a semisimple monoidal Hom-Hopf algebra (H, α). Finally, we form an associated Morita context [AbiH, AbiHAA☆H, A☆HAAbiH, A☆H] in the category .

MORITA CONTEXT FOR WEAK DOI-KOPPINEN SMASH PRODUCTS AND ITS APPLICATIONS

The paper is concerned with the Morita context for weak Doi-Koppinen smash products and the surjectivity of Morita maps are studied. As an application, we obtain a Morita context for endomorphism algebras for weak Doi-Hopf modules induced by coquasitriangular weak Hopf algebras.

Globalization of partial actions of groupoids on nonunital rings

We prove a globalization theorem for a partial action of a groupoid on a [Formula: see text]-unital ring. We also construct a strict Morita context between the partial skew groupoid ring and the skew groupoid ring associated to the globalization.

Morita equivalence based on Morita context for arbitrary semigroups

In this paper, we study the Morita context for arbitrary semigroups. We prove that, for two semigroups S and T, if there exists a Morita context $(S, T, P, Q, \tau, \mu)$ (not necessary unital) such that the maps $\tau$ and $\mu$ are surjective, the categories U S -FAct and U T -FAct are equivalent. Using this result, we generalize Theorem 2 in [2] to arbitrary semigroups. Finally, we give a characterization of Morita context for semigroups.

Semicommutativity of the rings relative to prime radical

In this paper, we introduce a new kind of rings that behave like semicommutative rings, but satisfy yet more known results. This kind of rings is called $P$-semicommutative. We prove that a ring $R$ is $P$-semicommutative if and only if $R[x]$ is $P$-semicommutative if and only if $R[x, x^{-1}]$ is $P$-semicommutative. Also, if $R[[x]]$ is $P$-semicommutative, then $R$ is $P$-semicommutative. The converse holds provided that $P(R)$ is nilpotent and $R$ is power serieswise Armendariz. For each positive integer $n$, $R$ is $P$-semicommutative if and only if $T_n(R)$ is $P$-semicommutative. For a ring $R$ of bounded index $2$ and a central nilpotent element $s$, $R$ is $P$-semicommutative if and only if $K_s(R)$ is $P$-semicommutative. If $T$ is the ring of a Morita context $(A,B,M,N,\psi,\varphi)$ with zero pairings, then $T$ is $P$-semicommutative if and only if $A$ and $B$ are $P$-semicommutative. Many classes of such rings are constructed as well. We also show that the notions of clean rings and exchange rings coincide for $P$-semicommutative rings.

Morita, Galois, and the trace map: a survey

Let A be a \({\mathcal {K}}\)-algebra and H a \({\mathcal {K}}\)-bialgebra (\({\mathcal {K}}\) being a field). Any action \(\beta \) of H on A gives rise to two new \({\mathcal {K}}\)-algebras, namely, the algebra \(A^\beta \) of the invariants of A under \(\beta \) and the smash product \(A\#_\beta H\), as well as a canonical Morita context connecting them. Such a context keeps a close relation with the notion of Galois extension. Indeed, in some cases where it makes sense the strictness of this context is equivalent to exactly say that A is a \(H^*\)-Galois extension of \(A^\beta \). In general, such an equivalence depends also on the surjectivity of a certain trace map from A to \(A^\beta \). This paper is a survey about the strictness of this context in the setting of partial actions of groups and of Hopf algebras.

Morita Equivalence of Semirings and Its Connection with Nobusawa Γ-Semirings with Unities

We show that the left operator semiring and the right operator semiring of a Nobusawa Γ-semiring with unities are Morita equivalent. The converse is also deduced, i.e., for two Morita equivalent semirings L and R, a Nobusawa Γ-semiring A with unities is constructed such that the left and right operator semirings of A are isomorphic to L and R, respectively. As an application we first establish a relationship between Morita equivalence and Morita context of semirings, and then enumerate some properties of semirings which remain invariant under Morita equivalence.

Partial actions of weak Hopf algebras: Smash product, globalization and Morita theory

In this paper we introduce the notion of partial action of a weak Hopf algebra on algebras, unifying the notions of partial group action [11], partial Hopf action [2,3,9] and partial groupoid action [4]. We construct the fundamental tools to develop this new subject, namely, the partial smash product and the globalization of a partial action, as well as we establish a connection between partial and global smash products via the construction of a surjective Morita context. In particular, in the case that the globalization is unital, these smash products are Morita equivalent. We show that there is a bijective correspondence between globalizable partial groupoid actions and symmetric partial groupoid algebra actions, extending similar result for group actions [9]. Moreover, as an application we give a complete description of all partial actions of a weak Hopf algebra on its ground field, which suggests a method to construct more general examples.

The p.p. property of trivial extensions

In this note, a necessary and sufficient condition for a trivial extension to be a left p.p.-ring is obtained. As an application, we determine when a trivial Morita context is a left p.p.-ring, and a result of Xue [On p.p. rings, Kobe J. Math. 7 (1990) 77–80] characterizing triangular p.p.-rings follows immediately.

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.

A note on Morita invariants of semigroups

In this paper we observe that important ideals like prime ideals, nil ideals, nilpotent ideals, primary ideals etc. remain invariant under strong Morita equivalence of semigroups with weak local units. As a consequence, some radicals and important properties are found to be invariant. Finally, lattice isomorphisms between the set of all ideals and the set of all sub-biacts corresponding to a Morita context of the said class of semigroups are established.

BAER SPECIAL RINGS AND REVERSIBILITY

Abstract. In this paper, we apply some properties of reversiblerings, Baerness of xed rings, skew group rings and Morita Contextrings to get conditions that shows xed rings, skew group ringsand Morita Context rings are reversible. Moreover, we investigateconditions in which Baer rings are reversible and reversible ringsare Baer. 1. IntroductionThroughout this paper all rings are associative rings with identity un-less otherwise stated. Let Rbe a ring. We denote S r (R)(resp. S l (R) bythe set of right(resp. left)semi-central idempotents in R. For a nonemptysubset Xof R, r R (X)(resp. l R (X)) will be denoted by the right(resp.left)annihilator of X in R. In 1990, Habeb studied commutativering in [5]. A ring R is called commutative, if ab = 0 impliesba= 0 for any a;b2R. In 1999[4] used the terminology reversiblering instead of zero commutative . Obviously, a commutative ringis reversible ring, but converse is not true. In fact every reversible ringis semi-commutative but converse is not true(for more details see [12]).Moreover, A ring Ris called right (resp. left)symmetric ring, if rst= 0implies rts= 0 (resp. srt= 0), for any r;s;t2R. It is easy to checkthat reduced ring is symmetric ring,a symmetric ring with identity isa reversible and a reversible ring is semi-commutative. In 2002, Marks[16] studied conditions in which a group ring becomes reversible, andstudied some relationships of among symmetric rings, reduced rings andreversible rings. Baer ring is one of the classic rings and it is appliedwidely in the eld of C*-algebra, Von Neumann algebra and Coding

On generalized partial twisted smash products

We first introduce the notion of a right generalized partial smash product and explore some properties of such partial smash product, and consider some examples. Furthermore, we introduce the notion of a generalized partial twisted smash product and discuss a necessary condition under which such partial smash product forms a Hopf algebra. Based on these notions and properties, we construct a Morita context for partial coactions of a co-Frobenius Hopf algebra.

Centralizing mappings, Morita context and generalized (α, β)-derivations

In the present note, we use a pair of rings, which are the ingredients of a Morita context, and obtain that if one of the ring is prime with the generalized (α, β)-derivations that satisfy certain conditions on the trace ideal of the ring, which by default is a Lie ideal, and the other ring is reduced, then the trace ideal of the reduced ring is contained in the center of the ring. As an outcome, in case of a semi-projective Morita context, the reduced ring becomes commutative.

Line bundles and the Thom construction in noncommutative geometry

The idea of a line bundle in classical geometry is transferred to noncommutative geometry by the idea of a Morita context. From this we construct \mathbb{Z} - and \mathbb{N} -graded algebras, the \mathbb{Z} -graded algebra being a Hopf–Galois extension. A non-degenerate Hermitian metric gives a star structure on this algebra, and an additional star operation on the line bundle gives a star operation on the \mathbb{N} -graded algebra. In this case, we carry out the associated circle bundle and Thom constructions. Starting with a C*-algebra as base, and with some positivity assumptions, the associated circle and Thom algebras are also C*-algebras. We conclude by examining covariant derivatives and Chern classes on line bundles after the method of Kobayashi and Nomizu.

SMASH PRODUCTS, SEPARABLE EXTENSIONS AND A MORITA CONTEXT OVER HOPF ALGEBROIDS

Let [Formula: see text] be a Hopf algebroid, and A a left [Formula: see text]-module algebra. This paper is concerned with the smash product algebra A#H over Hopf algebroids. In this paper, we investigate separable extensions for module algebras over Hopf algebroids. As an application, we obtain a Maschke-type theorem for A#H-modules over Hopf algebroids, which generalizes the corresponding result given by Cohen and Fischman in [Hopf algebra actions, J. Algebra100 (1986) 363–379]. Furthermore, based on the work of Kadison and Szlachányi in [Bialgebroid actions on depth two extensions and duality, Adv. Math.179 (2003) 75–121], we construct a Morita context connecting A#H and [Formula: see text] the invariant subalgebra of [Formula: see text] on A.

Morita context, partial Hopf Galois extensions and partial entwining structure

We first introduce the notion of a right generalized partial smash product and explore some properties of such partial smash product. Based on these notions and properties, then we construct a Morita context for partial coactions of a co-Frobenius Hopf algebra. Finally, we prove that any Hopf partial Galois extension induces a unique partial entwining map compatible with the right partial coaction.

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.