Semigroups strongly Morita equivalent to monoids

Periodica Mathematica Hungarica - We consider semigroups strongly Morita equivalent to a fixed monoid. We prove that such semigroups are precisely the enlargements of that monoid. We also show that...     

Morita duality and graded rings

ABSTRACT

We find representatives of all the equivalence classes of simple root systems (or r.e.s. for brevity) for the complex reflection groups G ₁₂ , G ₂₄ , G ₂₅ and G ₂₆ . Then we give representatives of all the congruence classes of (essential) presentations (or r.c.p. (r.c.e.p.) for brevity) for these groups by generators and relations. The method used in the paper is applicable to any finite (complex) reflection groups.     

Co-Artinian rings and Morita duality

A ringA is left co-Noetherian if the injective hull of each simple leftA-module is Artinian. Such rings have been studied by Vámos and Jans. Dually, callA left co-Artinian if the injective hull of each simple leftA-module is Noetherian. Left co-Artinian rings having only finitely many nonisomorphic simple left modules are studied, and such rings are shown to have nilpotent radical. Moreover, it is shown that left co-Artinian implies left co-Noetherian ifA/J is Artinian. For an injective leftA-module _A Q withB=End ( _A Q), andC=End (Q _B ), conditions yielding a Morita duality between and are obtained. In special cases, e.g. _A Q a self-cogenerator, this Morita duality yields chain conditions on _A Q. Specialized to commutative rings, these results give the known fact that every commutative co-Artinian ring is co-Noetherian. Finally in the case that the injective hull _A E=E( _A S) of a simple leftA-module _A S is a self-cogenerator, chain conditions on _A E are related to chain conditions onB _B =End ( _A E). The results obtained are analogous to results for commutative rings of Vámos, Rosenberg and Zelinsky. It is shown that ifA is a left co-Artinian ring withE( _A S) a self-cogenerator for each simple _A S, thenJ is nil and .     

Artinian rings with Morita duality

Two theorems are established which properly extend the ckss of artinian rings with Morita duality. A question of Anh about QF-rings is answered in the negative. Project supported by the National Natural Science Foundation of China (Grant No. 19771016).     

Morita duality emerging from quasi-abelian categories

Algebras and Representation Theory - It is shown that a general concept of Morita duality between abelian categories with no generating hypothesis for reflexive objects is completely described by a...     

Endomorphism monoids of acts over monoids

In this paper we investigate under which conditions a monoid R is defined by the endomorphism monoid of an act over R. More precisely, we ask when an isomorphism between two such endomorphism monoids over monoids R₁ and R₂ is induced by a semilinear isomorphism. The question is considered also for ordered and for topological monoids. On the way we characterize monoids over which all projective acts are free. An abstract of this paper appeared in the Proceedings of the Conference on Semigroups, Szeged 1972.     

Constructions and presentations for monoids

Presentations are found for the wreath product of two monoids, the Schützenberger product of two monoids, the Bruck-Reilly extension of a monoid, strong semilattices of monoids and Rees matrix semigroups of monoids.     

可分解半群的Morita等价

设S,R是可分解半群.记US-FAct={sM∈S-Act|SM=M且SHoms(S,M)≌M],给出了范畴US-FAct与UR-FAct等价的刻划;S分别强Morita等价于一个夹层半群、局部单位半群、幺半群和群的条件;S是完全单半群当且仅当S强Morita等价于一个群且对任何指标集I,S SHoms(S,i∈I S)→i∈I S,s t·f→(st)f,是同构.     

Ideals in Morita Rings and Morita Semigroups

We characterize the lattice of all ideals of a Morita ring （semigroup） when the corresponding pair of rings （semigroups） in the Morita context are Morita equivalent s-unital （like-unitv） rings （semigroups）.     

Homotopical Morita theory for corings

A coring (A,C) consists of an algebra A in a symmetric monoidal category and a coalgebra C in the monoidal category of A-bimodules. Corings and their comodules arise naturally in the study of Hopf–Galois extensions and descent theory, as well as in the study of Hopf algebroids. In this paper, we address the question of when two corings (A,C) and (B,D) in a symmetric monoidal model category V are homotopically Morita equivalent, i.e., when their respective categories of comodules V _A ^C and V _B ^D are Quillen equivalent. As an illustration of the general theory, we examine homotopical Morita theory for corings in the category of chain complexes over a commutative ring.     

Morita Context环的幂零性

设(A,B,V,W,(),[])是一个Morita Context,C=A VW B是对应的Morita Context环.用基本环论方法,给出了C与A,B,V,W之间关于环的诣零性,幂零性,局部幂零性,N—诣零性,P—性等性质的关系.     

Morita Equivalence for Factorisable Semigroups

Recall that the semigroups S and R are said to be strongly Morita equivalent if there exists a unitary Morita context (S, R., _S P _R,R Q _S ,〈〉 , ⌈⌉) with 〈〉 and ⌈⌉ surjective. For a factorisable semigroup S, we denote ζ _S = {(s ₁, s ₂) ∈S×S|ss ₁ = ss ₂, ∀s∈S}, S' = S/ζ _S and US-FAct = { _S M∈S− Act |SM = M and SHom _S (S, M) ≅M}. We show that, for factorisable semigroups S and M, the categories US-FAct and UR-FAct are equivalent if and only if the semigroups S' and R' are strongly Morita equivalent. Some conditions for a factorisable semigroups to be strongly Morita equivalent to a sandwich semigroup, local units semigroup, monoid and group separately are also given. Moreover, we show that a semigroup S is completely simple if and only if S is strongly Morita equivalent to a group and for any index set I, S⊗SHom _S (S, ∐ _i∈I S) →∐ _i∈I S, s⊗t·ƒ↦ (st)ƒ is an S-isomorphism. The research is partially supported by a UGC(HK) grant #2160092. Project is supported by the National Natural Science Foundation of China     

Right-orderability versus left-orderability for monoids

Semigroup Forum - We investigate the relationship between (total) left- and right-orderability for monoids, in particular illustrating the finite case by various structural observations and...     

Morita Invariance for Infinitesimal Deformations

Algebras and Representation Theory - Let A and B be two Morita equivalent finite dimensional associative algebras over a field ��. It is well known that Hochschild cohomology is...