bin Li

Quasi-Normal Rings

A ring R is defined to be quasi-normal if ae = 0 implies eaRe = 0 for a ∈ N(R) and e ∈ E(R), where E(R) and N(R) stand, respectively, for the set of idempotents and the set of nilpotents of R. It is proved that R is quasi-normal if and only if eR(1 − e)Re = 0 for each e ∈ E(R) if and only if T n (R, R) is quasi-normal for any positive integer n. And it follows that for a quasi-normal ring R, R is Π-regular if and only if N(R) is an ideal of R and R/N(R) is regular. Also, using quasi-normal ring, we proved the following: (1) R is an abelian ring if and only if R is a quasi-normal left idempotent reflexive ring; (2) R is a strongly regular ring if and only if R is a von Neumann regular quasi-normal ring; (3) Let R be a quasi-normal ring. Then R is a clean ring if and only if R is an exchange ring; (4) Let R be a quasi-normal Π-regular ring. Then R is a (S,2)-ring if and only if ℤ/2ℤ is not a homomorphic image of R.

WEIGHT PROPERTY FOR IDEALS OF U q (sl(2))

Let k be an arbitrary field of characteristic zero, and U be the quantized enveloping algebra U q (sl(2)) over k. The aim of this present paper is to study the ideals of U at q not a root of unity. It turns out that every non-zero ideal of U can be generated by at most two highest weight vectors under the adjoint action, and by a sum of two highest weight vectors. This weight property make it possible to give a complete list of all prime (primitive, maximal) ideals of U according to their generators.

MC2 RINGS AND WQD RINGS

We introduce in this paper the concepts of rings characterized by minimal one-sided ideals and concern ourselves with rings containing an injective maximal left ideal. Some known results for idempotent reflexive rings and left HI rings can be extended to left MC2 rings. As applications, we are able to give some new characterizations of regular left self-injective rings with non-zero socle and extend some known results for strongly regular rings.

Self-Dual Hopf Quivers

ABSTRACT We study self-dual coradically graded pointed Hopf algebras with a help of the dual Gabriel theorem for pointed Hopf algebras (van Oystaeyen and Zhang, 2004). The co-Gabriel Quivers of such Hopf algebras are said to be self-dual. An explicit classification of self-dual Hopf quivers is obtained. We also prove that finite dimensional pointed Hopf algebras with self-dual graded versions are generated by group-like and skew-primitive elements as associative algebras. This partially justifies a conjecture of Andruskiewitsch and Schneider (2000) and may help to classify finite dimensional self-dual coradically graded pointed Hopf algebras.

A CRITERION FOR THE JACOBSON SEMISIMPLICITY OF THE GREEN RING OF A FINITE TENSOR CATEGORY

The first author was funded by China Postdoctoral Science Foundation (Grant No. 2017M610316), the second author was funded by the NSF of China (Grant No. 11471282).