Throughout this paper , we assume that a and b are rings with identities and the only idempotents in a and b are 1 and 0 设a ， b是两个有单位元的环，并且他们都只有平凡的幂等元， m为非零的( a ， b ) -双模。
It is proved that every nonzero ideal in a finite - dimensional semi - simple algebra over a field is generated by an unique central idempotent 证明了域上有限维半单代数的每一个非零理想由唯一的中心幂等元生成。
In the study of semigroup theory . a very basic problem is that what kind of the semigroup ' s property we can get from its idempotent ' s property 在半群代数理论中，一个非常基本的问题是，从半群的幂等元的性质可以得到该半群的何种性质
Definition 1 let s be e - inversive semigroup . then s is called a weak r - uuipotent e - iiiversivc semigroup if it satisfies t he following definition 2 let p he a congruence on s ( r ) . thei ) p is a r - congruence if it satisfies the following main result 3 then t is the maximum idempotent separating r - congruence on a weak r - 接着给出s田）上的最大的幂等元分离r同余的一个刻画，介绍r正则、 r正规子集的概念最后给出s叮）上的幂等元分离r同余格的一个刻画
As a both generalization of orthodox semigroup and regular * - semigroup , ya - mada and sen gave the concept of ( weakly ) p - regular semigroup in [ 5 ] , about the p - regular semigroup and ( weakly ) p - regular semigroup , sen ^ ^ seth and rui changxiang m have given a equivalent characterization of idempotet separating congruence lattice on them separately . as a generalize of this , in the forth section , the concept of p - separating congruence on a regualr ^ - semigroup is introduced . and we discuss some properties about it . after introducing the concept of p - normal subset of a regualr * - semigroup , we discribe the lattice of p - separating congruences contained in green ' s relations ? k ) in regular semigroup 和seth及芮昌祥间分别给出了其上幂等元分离同余格的一个等价刻画作为对此的一个推广，在第四节中，我们首先引入正则”半群上r分离同余的概念，然后研究其性质和等价刻画，最后给出包含在二仰）中p分离同余格的一个等价刻画。定义二：令s为正则”半群， p为s上的”同余
