Morphic property of a quotient ring over polynomial ring
Bull. Korean Math. Soc. 2013 Vol. 50, No. 5, 1433-1439
Published online September 1, 2013
Kai Long, Qichuan Wang, and Lianggui Feng
Aviation University of Air Force, National University of Defense Technology
Abstract : A ring $R$ is called \textit{left morphic} if $R/Ra\cong l(a)$ for every $a\in R$. Equivalently, for every $a\in R$ there exists $b\in R$ such that $Ra=l(b)$ and $l(a)=Rb$. A ring $R$ is called \textit{left quasi-morphic} if there exist $b$ and $c$ in $R$ such that $Ra=l(b)$ and $l(a)=Rc$ for every $a\in R$. A result of T.-K. Lee and Y. Zhou says that $R$ is unit regular if and only if $R[x]/(x^2)\cong R\propto R$ is morphic. Motivated by this result, we investigate the morphic property of the ring $S_n\stackrel{\mathrm{def}}{=}R[x_1,x_2,\dots,x_n]/(\{x_ix_j\})$, where $i, j\in \{1,2,\dots,n\}$. The morphic elements of $S_n$ are completely determined when $R$ is strongly regular.
Keywords : morphic property, polynimial ring, strongly regular
MSC numbers : 16E50, 13F20, 16U99
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