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\section*{On the problem of diagonalizability of matrices over a principal ideal domain} By {\sl Volodymyr Prokip}.  

\medskip  \noindent  Let  $R$ -- be a principal ideal domain with the unit element $e\not=0$ and $U(R)$ the set of divisors of unit element $e$. Further, let $R_n$ -- the ring of $(n\times n)$-matrices over $R$; $I_k$ -- the identity $k\times k$ matrix and $O$ the zero $n\times n$ matrix.  In this report we present conditions of diagonalizability of a matrix $A \in R_n$, i.e. when for $A$ there exists  a matrix  $T \in GL(n,R)$ such that $TAT^{-1}$ --  a diagonal matrix.   

{\bf Theorem.}  Let $A\in R_n$ and $$\det (Ix-A)=(x-\alpha_1)^{k_1}(x-\alpha_2)^{k_2}       \cdots (x-\alpha_r)^{k_r} , $$ where $ \alpha_i \in R $, and $ \alpha_i - \alpha_j \in U(R)$ for all $i\not= j$. If $m(x)=(x-\alpha_1)(x-\alpha_2) \cdots (x-\alpha_r)$ -- the minimal polynomial of the matrix $A$, i.e. $m(A)=O$, then for the matrix  $A$ there exists a matrix  $ T \in GL(n,R)$  such that $$    TAT^{-1}={\rm diag} \left( {\alpha}_1I_{k_1}, {\alpha}_2I_{k_2},       \ldots , {\alpha}_rI_{k_r} \right) .  $$
%matrix , pricipal ideal domain, diagonalization","15A04","15A21","","11:23:57","Mon Mar 31 2008","194.44.153.33"
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