ra.rings and algebras - For every ring R, is there a block-diagonal canonical form for a square matrix over R? - MathOverflow

Question

This question asks whether there exists an analogue of the Jordan decomposition for an arbitrary ring $R$. This analogue is not necessarily the Jordan-Chevalley decomposition, which is unnecessarily strong. This follows from this question, but you don't need to read it.

Given a ring $R$, let $J(R)$ be the monoid whose elements are all square matrices over $R$, and where the monoid operation is $oplus$ denoting direct sum of matrices. Let $A { sim_ ext{S}} B$ mean that there exists an invertible matrix $P$ such that $PAP^{-1} = B$. Is the monoid $J(R)/{ sim_ ext{S}}$ always a free abelian monoid?

For perfect fields, the answer is yes by the Jordan-Chevalley decomposition. What about for every ring? Is there a counterexample?

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Answer 1

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