The Source of Semiprimeness of Rings
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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.
Abstract
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.
참고문헌 (3)
- T. Y. Lam, "A First Course in Noncommutative Rings" (1991), Springer-Verlag
- T. Y. Lam, "Lectures on Modules and Rings" (1999), Springer-Verlag
- N. H. McCoy, "The Theory of Rings" (1964), The Macmillan Co.