J. Korean Math. Soc. 2014; 51(4): 751-771
Printed July 1, 2014
https://doi.org/10.4134/JKMS.2014.51.4.751
Copyright © The Korean Mathematical Society.
Nam Kyun Kim, Yang Lee, and Yeonsook Seo
Hanbat National University, Pusan National University, Pusan National University
We study the structure of idempotents in polynomial rings, power series rings, concentrating in the case of rings without identity. In the procedure we introduce right Insertion-of-Idempotents-Property (simply, right IIP) and right Idempotent-Reversible (simply, right IR) as generalizations of Abelian rings. It is proved that these two ring properties pass to power series rings and polynomial rings. It is also shown that $\pi$-regular rings are strongly $\pi$-regular when they are right IIP or right IR. Next the noncommutative right IR rings, right IIP rings, and Abelian rings of minimal order are completely determined up to isomorphism. These results lead to methods to constructsuch kinds of noncommutative rings appropriate for the situations occurred naturally in studying standard ring theoretic properties.
Keywords: idempotent, right IIP ring, right IR ring, Abelian ring
MSC numbers: 16P10, 16U80
2016; 53(1): 217-232
2017; 54(1): 177-191
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