标题： 关于广义Rickart$*$-环 显示英文标题

标题： On Generalized Rickart $*$-rings

摘要： 一个带有对合$*$的环$R$是广义 Rickart$*$-环，如果对于所有$x\in R$，$x^n$的右零化子由某个与$x$有关的正整数$n$生成的投影所生成。 在本文中，我们引入了$*$-环中一个元素的广义右投影，并证明了广义Rickart$*$-环中的每个元素都有广义右投影。 获得了广义Rickart$*$-环的各种特征。 我们引入了广义弱Rickart$*$-环的概念，并以弱广义Rickart$*$-环来表征广义Rickart$*$-环。 表明广义Rickart$*$-环满足平行四边形法则。 对于广义Rickart$*$-环中的部分可比较性，建立了充分条件。 此外，证明了广义Rickart$*$-环中的一对投影具有正交分解。 显示英文摘要

摘要： A ring $R$ with an involution $*$ is a generalized Rickart $*$-ring if for all $x\in R$ the right annihilator of $x^n$ is generated by a projection for some positive integer $n$ depending on $x$. In this work, we introduce generalized right projection of an element in a $*$-ring and prove that every element in a generalized Rickart $*$-ring has generalized right projection. Various characterizations of generalized Rickart $*$-rings are obtained. We introduce the concept of generalized weakly Rickart $*$-ring and provide a characterization of generalized Rickart $*$-rings in terms of weakly generalized Rickart $*$-rings. It is shown that generalized Rickart $*$-rings satisfy the parallelogram law. A sufficient condition is established for partial comparability in generalized Rickart $*$-rings. Furthermore, it is proved that pair of projections in a generalized Rickart $*$-ring possess orthogonal decomposition.