Elementary divisor theorem for noncommutative PIDs

Guralnick, Robert M.; Levy, Lawrence S.; Odenthal, Charles

doi:10.1090/S0002-9939-1988-0954973-3

Elementary divisor theorem for noncommutative PIDs
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by Robert M. Guralnick, Lawrence S. Levy and Charles Odenthal PDF

Proc. Amer. Math. Soc. 103 (1988), 1003-1011 Request permission

Abstract:

We prove that, over a PID, if two matrices ${\mathbf {A}}$ and ${\mathbf {B}}$ have the same size, present isomorphic modules and have rank $\geq 2$, then ${\mathbf {A}}$ is equivalent to ${\mathbf {B}}$. This answers a question raised by Nakayama in 1938. Our solution makes use of a number of facts about the algebraic $K$-theory of noetherian rings.

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