Diagonalizability of idempotent matrices over integral domains

Determine whether every idempotent matrix E in M_n(R) over an integral domain R is diagonalizable over R in the sense of ring-similarity; that is, ascertain whether there always exists an invertible matrix A in GL_n(R) such that E = A diag(I_{rank(E)}, 0) A^{-1}.

Background

Over division rings, every idempotent matrix is similar to a block diagonal matrix with an identity block and a zero block, which underpins the poset analysis developed earlier in the paper. Extending this structural property beyond division rings is nontrivial because standard polynomial techniques do not directly apply in general rings.

The authors focus on principal ideal domains (PIDs) using Smith normal form to obtain constructive criteria for idempotents, and they formulate a concrete PID question. However, they explicitly acknowledge that, in general over integral domains, it is unknown whether idempotent matrices admit such a diagonalization by a matrix invertible over the domain.