Commun. Korean Math. Soc. 2018; 33(4): 1083-1096
Online first article March 26, 2018 Printed October 31, 2018
https://doi.org/10.4134/CKMS.c170409
Copyright © The Korean Mathematical Society.
Ne\c{s}et Aydin, \c{C}a\u{g}r\i\ Dem\.ir, D\.{i}dem Karalarl\i o\u{g}lu Camc\i
\c{C}anakkale Onsek\.{I}z Mart University, Ege University, \c{C}anakkale Onsek\.{I}z Mart University
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
2023; 38(1): 79-87
2022; 37(2): 359-370
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