MP-injective rings and MGP-injective rings

Abstract

A ring R is said to be right MP-injective if every monomorphism from a principal right ideal to R extends to an endomorphism of R. A ring R is said to be right MGP-injective if, for any 0 = a ∈ R, there exists a positive integer n such that a n = 0 and every monomorphism from a n R to R extends to R. We shall study characterizations and properties of these two classes of rings. Some interesting results on these rings are obtained. In particular, conditions under which right MGP-injective rings are semisimple artinian rings, von Neumann regular rings, and QF-rings are given.

Facile prepared bis(carbazyl thiocarbonyl) disulfide as chain transfer agent for RAFT polymerization of methyl methacrylate

“…Increasing the BCTD concentration would increase the probability of the reaction between BCTD and AIBN, which can decrease the concentration of the propagating radical and eventually lead to decreasing the polymerization rate. These results are in agreement with the previous work of Monteiro and de Brouwer,26 and Zhou and coworkers,21 in which bis(benzyl thiocarbonyl) disulfide and bis(thionaphthoyl) disulfide were used to in situ produce RAFT agent, respectively. In the same way, the reaction possibility of AIBN and BCTD would increase when increasing the AIBN concentration while keeping BCTD concentration constant.…”

“…These efficient CATs are crucial for the RAFT process of some less reactive monomers, for example, methyl methacrylate (MMA), so it is important to find a stable, easily synthetic route, or other alternative agents 19, 20. Some groups successively reported the RAFT homopolymerizations of styrene and (meth)acrylate monomers by using bis(thiobenzoyl) disulfide and azobis(isobutyronitrile) (AIBN) to synthesize CTA in situ 21, 22. Compared with dithioesters, the bis(dithiocarbonyl) disulfide is much stable and easier to prepare and purify.…”

Almost MGP-Injective Rings

“….} ⊆ R, by [7] (Lemma 3.10), Z r is right T -nilpotent, and so Z r is nil. It follows that Z r ⊆ J(R), and hence J(R) = Z r .…”

“…By Theorem 2.3, R is right perfect, i.e., R/J(R) is semisimple Artinian and J(R) is right T − nilpotent. If J(R) = 0, then, by [7] (Lemma 3.16), J(R) is not nil, a contradiction. So J(R) = 0, and whence R is semisimple Artinian.…”

“…In [7], the concepts of right P-injective rings and right GP-injective rings are generalized to right MP-injective rings and right MGP-injective rings, respectively. Following [7], a ring R is called right MP-injective if, for every R-monomorphism from a principal right ideal of R to R extends to an endomorphism of R; a ring R is called right MGP-injective if, for any 0 = a ∈ R, there exists a positive integer n such that a n = 0 and any R-monomorphism from a n R to R extends to a endomorphism of R. In this paper, we shall generalize the concept of right MGP-injective rings to right AMGP-injective rings, some properties of these rings will be given, conditions under which right AMGP-injective rings are QF-rings, semisimple Artinian rings and simple Artinian rings will be given, respectively. And right AMGP-injective left Noertherian rings will be investigated.…”