On co-semihereditary rings

A ringR is left co-semihereditary (strongly left co-semihereditary) if every finitely cogenerated factor of a finitely cogenerated (arbitrary) injective leftR-module is injective. A left co-semihereditary ring, which is not strongly left co-semihereditary, is given to answer a question of Miller and Tumidge in the negative. If _R U _S defines a Morita duality,R is proved to be left co-semihereditary (left semihereditmy) if and only ifS is right semihereditary (right co-semihereditary). Assuming thatS⩾R is an almost excellent extension,S is shown to be (strongly) right co-semihereditary if and only ifR is (strongly) right co-semihereditary. Project supported by the National Natural Science Foundation of China.     

Matrix Rings with Summand Intersection Property

A ring R has right SIP (SSP) if the intersection (sum) of two direct summands of R is also a direct summand. We show that the right SIP (SSP) is the Morita invariant property. We also prove that the trivial extension of R by M has SIP if and only if R has SIP and (1 – e)Me = 0 for every idempotent e in R. Moreover, we give necessary and sufficient conditions for the generalized upper triangular matrix rings to have SIP.     

模糊概念格与模糊推理

基于伽罗瓦连接,分别在交换伴随对与对合剩余格条件下,讨论了模糊概念格的四种定义形式。并证明了在对合剩余格上,对偶性成立,四种模糊算子将具有与经典意义下一致的相互关系。最后我们提出了一种基于模糊概念格的模糊推理规则,并证明了其还原性。     

Kac-系统与Morita等价

本文讨论与Hilbert C~*-模相关的Kac-系统在C~*-代数上的作用的性质,给出了两个Morita等价的C~*-代数(在Kac-系统的作用下得到的C~*-代数的余交叉积是Morita等价的)。从而,Baaj和Skandalis等人的结论是本文定理的特殊情况。     

具有一对零态射的Morita Context环(II)

设$(A,B,V,W,\psi,\phi)$是一个Morita Context,具有一对零态射$\psi=0$, $\phi=0$, $C =\left ( \begin{array} {cc}A & V \\W & B \end{array}\right)$是对应的Morita Context环.本文给出了$C$与$A,B,V,W$之间关于环的$\pi$-正则性、semiclean性、Mophic性和环的Exchgange性、Potent性、GM性的关系.     

A remark on Mansfield's imprimitivity theorem

We show that the Morita equivalence part of Mansfield's Imprimitivity Theory can be obtained by Green's Imprimitivity Theorem (and duality theory).

     

α‐ZrP/Uracil/Cu2+ nanoparticles as an efficient catalyst in the Morita‐Baylis‐Hillman reaction

A facile synthesis of uracil‐Cu²⁺ nanoparticles immobilized on alpha‐zirconium hydrogen phosphate (α‐ZrP), abbreviated as α‐ZrP/Uracil/Cu²⁺, was presented. This compound was synthesized by the thermal method and used as a reusable catalyst for the Morita‐Baylis‐Hillman reaction without any additives. First, (3‐ iodopropyl) trimethoxysilane as a linker is reacted with α‐ZrP support to give the α‐ZrP/IPTMOS. Addition of uracil and then the addition of copper (II) acetate to α‐ZrP/IPTMOS results in the production of selected catalyst. The Morita‐Baylis‐Hillman reaction catalyzed by α‐ZrP/Uracil/Cu^{2 +} demonstrated high product yield, short reaction time and a straightforward work‐up. The catalyst with enough outside surface was easily recovered using centrifugation and reused five times without a significant reduction in its activity.     

Synthesis of multi‐functionalized carbonyl compounds by palladium–catalysed γ‐substitution of MBH adducts with activated amides

Catalytic allylic γ‐substitution with Morita‐Baylis‐Hillman (MBH) adducts for creating a new family of unsymmetrical dicarbonyl compounds was presented in this work, in which a variety of allylated amide products were achieved in good yields and high regioselectivity with excellent linear‐to‐branched ratios. Especially, it was found that the Pd/HZNU‐Phos complex exhibited remarkably high activity (with a TON up to 16800) in this transformation between dicarbonyl amides and MBH adducts. In addition, the possibly multisite interaction between multifunctional Pd/HZNU‐Phos catalyst system and substrates might responsible for its exceptionally high efficiency in this reaction.     

N‐Methylpiperidine—A Useful Base Catalyst in the Morita–Baylis–Hillman Reaction

N‐methylpiperidine, a commercially available mild base, has effectively been utilized as a catalyst in the Morita–Baylis–Hillman reaction. Moderate to excellent yields (34–95%) and significant rate enhancement have been observed.