I'm investigating $m \times n$ matrices over $R$, where $R$ is a finite commutative unitary ring, and more specifically a finite chain ring or a principal ideal ring. Define the row rank as the maximal number of lin. independent rows, the column rank in a similar manner, and the inner rank. My question is, are there necessary and sufficient conditions for $R$ such that the row rank is equal to the column rank, so that the term matrix rank is well-defined? I know that the two ranks coincide, for example, over a PID, and in this case they also coincide with other notions of rank I have encountered, such as the McCoy rank and the determinantal rank, but in general I don't know when I have the right to speak about the rank of a matrix. A reference would be welcome.
2026-03-31 13:33:25.1774964005
Over which rings is matrix rank defined?
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