Note on Morita Equivalence in Ring Extensions

It seems that Morita invariance judges of the importance of classes of ring extensions concerned. Miyashita introduced the notion of Morita equivalence in ring extensions, and he showed that the classes of G-Galois extensions and Frobenius extensions are Morita invariant. After that, Ikehata showed that the classes of separable extensions, Hirata separable extensions, symmetric extensions, and QF-extensions are Morita invariant. In this article, we shall prove that the classes of several extensions are Morita invariant. Further, we will give an example of the class of ring extensions which is not Morita invariant.     

Morita Invariance and Maximal Left Quotient Rings

Abstract

We prove that under conditions of regularity the maximal left quotient ring of a corner of a ring is the corner of the maximal left quotient ring. We show that if R and S are two non-unital Morita equivalent rings then their maximal left quotient rings are not necessarily Morita equivalent. This situation contrasts with the unital case. However we prove that the ideals generated by two Morita equivalent idempotent rings inside their own maximal left quotient rings are Morita equivalent.     

EXCHANGE MORITA RINGS

In this paper we characterize the largest exchange ideal of a ring R as the set of those elements x ∈ R such that the local ring of R at x is an exchange ring. We use this result to prove that if R and S are two rings for which there is a quasi-acceptable Morita context, then R is an exchange ring if and only if S is an exchange ring, extending an analogue result given previously by Ara and the second and third authors for idempotent rings. We introduce the notion of exchange associative pair and obtain some results connecting the exchange property and the possibility of lifting idempotents modulo left ideals. In particular we obtain that in any exchange ring, orthogonal von Neumann regular elements can be lifted modulo any one-sided ideal.     

Sesquilinear Morita Equivalence and Orthogonal Sum of Algebras with Antiautomorphism

We define a notion of Morita equivalence between algebras with antiautomorphisms such that two equivalent algebras have the same category of sesquilinear forms. This generalizes the Morita equivalence of algebras with involutions defined by Fröhlich and Mc Evett [5 Fröhlich , A. , McEvett , A. M. ( 1969 ). Forms over rings with involution . J. Algebra 12 : 79 – 104 .[Crossref], [Web of Science ®] , [Google Scholar]], and their categories of ?-hermitian forms.

For two Morita equivalent algebras with involution, with an additional technical property (which is true for central simple algebras), we define a new algebra with antiautomorphism, called the orthogonal sum, which generalizes the usual notion of orthogonal sum of forms. We explore the invariants of this sum.     

由双边群相关Hopf模诱导的Morita等价

Let H be a Hopf π-coalgebra over a commutative ring k with bijective antipode S, and A and B right π-H-comodulelike algebras. We show that the pair of adjoint functors （F3 = A  Bop A□ HBop -,G3 = （-）coH） between the categories A□HBopM and AMπB-H is a pair of inverse equivalences, when A is a faithfully flat π-H-Galois extension. Furthermore, the categories Moritaπ-H（A,B） and Morita □π-H（AcoH,BcoH） are equivalent, if A and B are faithfully flat π-H-Galois extensions.     

Switchable pyrrole-based hydrogen bonding motif in enantioselective trifunctional organocatalysis

Pyrroles are versatile chemical motifs for molecular recognition or ligand design but their utility as catalytic components are underexplored. We incorporated a pyrrole motif into our acid-switchable, MAP-based trifunctional organocatalytic system. The switching-on of this system by an external Brønsted acid, 3-methyl benzoic acid, presented proficient catalysis in both aza-Morita–Baylis–Hillman (MBH, up to >95% conversion and 88% ee over 3?h) and MBH (up to 81% conversion and 77% ee over 6?h) reactions. The enhanced catalytic generality and proficiency may be due to the new cooperativity via CH/π interactions between pyrrole and the acid switch.     

正则QB环上的Morita Contexts

假设T是Morita Contexts(A,B,M,N,φ,φ)的环,如果A和B是正则QB环,本文证明了T中任何元素都是幂等元和伪逆元之和.     

Crossed products for matrix rings

Let R be a ring and n≥2 an integer. We provide a systematic way to define new multiplications on M_n(R), the ring of n×n full matrices with entries in R. The obtained new rings Λ are Auslander–Gorenstein if and only if so is R.     

辛群胚的弱Morita等价与orbifold基本群

本文研究辛orbifold群胚的弱Morita等价,证明了两个辛orbifold群胚弱Morita等价当且仅当其orbifold基本群同构.