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%%% Primary Title: Morita equivalence lemma for arbitrary algebras
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%%% Filename: MoritaEquivalenceLemmaForArbitraryAlgebras.tex
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\begin{document}
\section{Morita equivalence lemma for arbitrary algebras}
Let us consider first an example of \htmladdnormallink{Morita equivalence}{http://planetphysics.us/encyclopedia/MoritaEquivalentAlgebras2.html}; thus, for an integer
$n \geq 1$, let $Mat_n(A)$ be the algebra of $n \times n$-matrices with entries in an algebra $A$. The following is a typical example of Morita equivalence
that involves \htmladdnormallink{noncommutative}{http://planetphysics.us/encyclopedia/AbelianCategory3.html} algebras.
\begin{theorem}{\bf Morita equivalence Lemma for arbitrary algebras}
For any algebra $A$ and any integer $n \geq 1$, the algebras $A$ and
$Mat_n(A)$ are Morita equivalent.
\end{theorem}
{\bf Important Notes:}
\begin{itemize}
\item Even if $A$ is a commutative algebra, the algebra $Mat_n(A)$ is
of course not commutative for any $n > 1$ because the \htmladdnormallink{matrix multiplication}{http://planetphysics.us/encyclopedia/Matrix.html} is generally \htmladdnormallink{non-commutative}{http://planetphysics.us/encyclopedia/AbelianCategory3.html}.
\item In general, the algebra $A$ cannot be recovered from its corresponding
\htmladdnormallink{abelian category}{http://planetphysics.us/encyclopedia/AbelianCategory2.html} $A$-mod. Therefore, in order for a \htmladdnormallink{concept}{http://planetphysics.us/encyclopedia/PreciseIdea.html} in \htmladdnormallink{noncommutative geometry}{http://planetphysics.us/encyclopedia/NoncommutativeGeometry4.html} to have or retain an intrinsic meaning, such a concept must be {\em Morita invariant} that is, to remain within the same Morita equivalence class.
This raises the important question: what properties of an algebra are Morita invariant ? The answer to this question is provided by the ``Uniqueness Morita
Theorem''.
\end{itemize}
\end{document}
