ABSTRACT

We study the one-sided version of central Drazin inverses. An element a in a ring R is said to be left central Drazin invertible if there is $x\in R$ such that $xa\in C\left(R\right)$, $x{a}^{n+1}=a^n$ for some $n\in \mathbb{N}$. The right central Drazin invertible elements can be defined similarly. It is shown that an element $a\in R$ is central Drazin invertible if and only if a is both left and right central Drazin invertible. Some well-known results on Drazin inverses and left invertible elements including the famous Kaplansky theorem in [Jacobson N. Some remarks on one-sided inverses. Proc Amer Math Soc. 1950;1:352–355] are generalized. As applications, we give a new characterization of Dedekind-finite rings from the point of view of one-sided central Drazin invertible elements. The Cline's formula on Drazin invertible elements is also generalized.

摘要

我们研究中央 Drazin 逆的单面版本。如果存在，则称环R中的元素a是左中心 Drazin 可逆的$X\in R$这样$X\mathrm{一种}\in C\left(R\right)$,$X{\mathrm{一种}}^{n+1}={\mathrm{一种}}^n$对于一些$n\in \mathbb{ñ}$. 可以类似地定义右中央 Drazin 可逆元素。表明一个元素$\mathrm{一种}\in R$是中心 Drazin 可逆的当且仅当a是左中心 Drazin 可逆和右中心 Drazin 可逆。关于 Drazin 逆和左可逆元素的一些著名结果，包括 [Jacobson N. 中的著名 Kaplansky 定理。关于单面逆的一些评论。Proc Amer 数学学会。1950;1:352–355] 被概括。作为应用，我们从单侧中心 Drazin 可逆元的角度给出了 Dedekind 有限环的新表征。Drazin 可逆元素的 Cline 公式也被推广。