16D40-PerfectAndSemiperfectRings.tex

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\pmtitle{perfect and semiperfect rings}
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\begin{document}
A ring $R$ is called \textbf{left/right perfect} if for any left/right $R$-module $M$ there exists a projective cover $p:P\to M$.

A ring $R$ is called \textbf{left/right semiperfect} if for any left/right finitely-generated $R$-module $M$ there exists a projective cover $p:P\to M$.

It can be shown that there are rings which are left perfect, but not right perfect. However being semiperfect is left-right symmetric property.

Some examples of semiperfect rings include:
\begin{enumerate}
\item perfect rings;
\item left/right Artinian rings;
\item finite-dimensional algebras over a field $k$.
\end{enumerate}
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