Elementary divisor theorem for noncommutative PIDs
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- by Robert M. Guralnick, Lawrence S. Levy and Charles Odenthal
- Proc. Amer. Math. Soc. 103 (1988), 1003-1011
- DOI: https://doi.org/10.1090/S0002-9939-1988-0954973-3
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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.References
- K. Asano, Nichtcommutative Hauptidealringe, Actualités. Sci. Indust., no. 696, Hermann, Paris, 1938.
- Andreas Dress, On the decomposition of modules, Bull. Amer. Math. Soc. 75 (1969), 984–986. MR 244227, DOI 10.1090/S0002-9904-1969-12326-8
- David Eisenbud and J. C. Robson, Modules over Dedekind prime rings, J. Algebra 16 (1970), 67–85. MR 289559, DOI 10.1016/0021-8693(70)90041-4
- Robert M. Guralnick, Matrix equivalence and isomorphism of modules, Linear Algebra Appl. 43 (1982), 125–136. MR 656440, DOI 10.1016/0024-3795(82)90248-8
- Robert M. Guralnick and Lawrence S. Levy, Presentations of modules when ideals need not be principal, Illinois J. Math. 32 (1988), no. 4, 593–653. MR 955382
- Nathan Jacobson, The Theory of Rings, American Mathematical Society Mathematical Surveys, Vol. II, American Mathematical Society, New York, 1943. MR 0008601
- Lawrence S. Levy and J. Chris Robson, Matrices and pairs of modules, J. Algebra 29 (1974), 427–454. MR 340337, DOI 10.1016/0021-8693(74)90079-9
- Tadasi Nakayama, A note on the elementary divisor theory in non-commutative domains, Bull. Amer. Math. Soc. 44 (1938), no. 10, 719–723. MR 1563855, DOI 10.1090/S0002-9904-1938-06850-4
- J. T. Stafford, Stable structure of noncommutative Noetherian rings, J. Algebra 47 (1977), no. 2, 244–267. MR 447335, DOI 10.1016/0021-8693(77)90224-1 O. Teichmüller, Der Elementarteilersatz für nichtkommutative Ringe, S.-B.-Preuss Akad. Wiss., 1937.
- R. B. Warfield Jr., Stable equivalence of matrices and resolutions, Comm. Algebra 6 (1978), no. 17, 1811–1828. MR 508083, DOI 10.1080/00927877808822323
Bibliographic Information
- © Copyright 1988 American Mathematical Society
- 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