Communications of the
Korean Mathematical Society CKMS

Let $R$ be an associative ring. We define a subset $S_{R}$ of $R$ as $S_{R}=\{a\in R \mid aRa=(0)\}$ and call it the \emph{source of semiprimeness of $R$}. We first examine some basic properties of the subset $S_{R}$ in any ring $R$, and then define the notions such as $R$ being a $|S_{R}|$-reduced ring, a $|S_{R}|$-domain and a $|S_{R}|$-division ring which are slight generalizations of their classical versions. Beside others, we for instance prove that a finite $|S_{R}|$-domain is necessarily unitary, and is in fact a $|S_{R}|$-division ring. However, we provide an example showing that a finite $|S_{R}|$-division ring does not need to be commutative. All possible values for characteristics of unitary $|S_{R}|$-reduced rings and $|S_{R}|$-domains are also determined.

Keywords: prime ideal, semiprime ideal, prime ring and semiprime ring

MSC numbers: Primary 16N60; Secondary 16U80, 16W25