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Proceedings of the American Mathematical Society

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Elementary divisor theorem for noncommutative PIDs


Authors: Robert M. Guralnick, Lawrence S. Levy and Charles Odenthal
Journal: Proc. Amer. Math. Soc. 103 (1988), 1003-1011
MSC: Primary 16A04; Secondary 16A14
DOI: https://doi.org/10.1090/S0002-9939-1988-0954973-3
MathSciNet review: 954973
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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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DOI: https://doi.org/10.1090/S0002-9939-1988-0954973-3
Article copyright: © Copyright 1988 American Mathematical Society