广义半交换环的幂零结构

Nilpotent Structure of Generalized Semicommutative Rings

中文关键词:  幂零$\alpha$-半交换环  $\alpha$-刚性环  $\alpha$-半交换环  多项式环

英文关键词:nilpotent $\alpha$-semicommutative rings  $\alpha$-rigid rings  $\alpha$-semicommutative rings  polynomial rings

中文摘要:

      研究了广义半交换环的幂零结构,定义了一类新的环类,即幂零$\alpha$-半交换环.说明了$\alpha$-半交换环与半交换环, $\alpha$-半交换环和$\alpha$-刚性环等环密切相关,通过构造反例说明了幂零$\alpha$-半交换环未必是$\alpha$-半交换环.研究了幂零$\alpha$-半交换环的各种性质,推广和统一了与环的半交换性质有关的若干结论.

英文摘要:

      We study the nilpotent structure of generalized semicommutative rings. The new concept of nilpotent $\alpha$-semicommutative rings is defined and studied. This class of rings is closely related to many well-known concepts including semicommutative rings, $\alpha$-semicommutative rings and weak $\alpha$-rigid rings. An example is given to show that a nilpotent $\alpha$-semicommutative ring need not be $\alpha$-semicommutative. Various properties of this class of rings are investigated. Many known results related to various semicommutative properties of rings are generalized and unified.