环中的中心clean元和中心Drazin逆

Centrally clean elements and central Drazin inverses in a ring

如果环R中元素a是一个中心幂等元e和一个可逆元u的和,那么称a是中心clean元,并且称a=u+e为a的一个中心clean分解.如果环R中所有元素都是中心clean元,则称环R是中心clean环.首先,给出了中心clean元的等价刻画,并进一步研究了中心clean环的一些性质以及环R是中心clean环的充分必要条件.中心clean环与中心Drazin逆有着紧密的联系.接着,从中心clean分解的角度给出了中心Drazin逆存在的充分必要条件,并研究了角环、幂级数环和环R_(α)的笛卡尔积∏R_(α)等特殊环的中心clean性.此外,还研究了一般域上代数中的2个中心幂等元组合的中心群可逆性.最后,作为一个应用,分别计算出了群环Z_(2)S_(3)中所有的可逆元、中心群可逆元、群可逆元、中心Drazin可逆元以及中心clean元.

Element a in ring R is called centrally clean if it is the sum of central idempotent e and unit u.Moreover,a=e+u is called a centrally clean decomposition of a and R is called a centrally clean ring if every element of R is centrally clean.First,some characterizations of centrally clean elements are given.Furthermore,some properties of centrally clean rings,as well as the necessary and sufficient conditions for R to be a centrally clean ring are investigated.Centrally clean rings are closely related to the central Drazin inverses.Then,in terms of centrally clean decomposition,the necessary and sufficient conditions for the existence of central Drazin inverses are presented.Moreover,the central cleanness of special rings,such as corner rings,the ring of formal power series over ring R,and a direct product ∏ R_(α) of ring R_(α),is analyzed.Furthermore,the central group invertibility of combinations of two central idempotents in the algebra over a field is investigated.Finally,as an application,an example that lists all invertible,central group invertible,group invertible,central Drazin invertible elements,and centrally clean elements of the group ring Z_(2)S_(3) is given.