J. Korean Math. Soc. 2010; 47(6): 1269-1282
Printed November 1, 2010
https://doi.org/10.4134/JKMS.2010.47.6.1269
Copyright © The Korean Mathematical Society.
Young Cheol Jeon, Hong Kee Kim, Nam Kyun Kim, Tai Keun Kwak, Yang Lee, and Dong Eun Yeo
Korea Science Academy, Gyeongsang National University, Hanbat National University, Daejin University, Busan National University, Busan National University
We in this note consider a new concept, so called $\pi$-McCoy, which unifies McCoy rings and IFP rings. The classes of McCoy rings and IFP rings do not contain full matrix rings and upper (lower) triangular matrix rings, but the class of $\pi$-McCoy rings contain upper (lower) triangular matrix rings and many kinds of full matrix rings. We first study the basic structure of $\pi$-McCoy rings, observing the relations among $\pi$-McCoy rings, Abelian rings, 2-primal rings, directly finite rings, and ($\pi$-)regular rings. It is proved that the $n$ by $n$ full matrix rings ($n\geq 2$) over reduced rings are not $\pi$-McCoy, finding $\pi$-McCoy matrix rings over non-reduced rings. It is shown that the $\pi$-McCoyness is preserved by polynomial rings (when they are of bounded index of nilpotency) and classical quotient rings. Several kinds of extensions of $\pi$-McCoy rings are also examined.
Keywords: $\pi$-McCoy ring, McCoy ring, polynomial ring, matrix ring, classical quotient ring
MSC numbers: 16N40, 16U80
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