Row and column finite matrices

Nielsen, Pace

doi:10.1090/S0002-9939-07-08790-4

Row and column finite matrices
HTML articles powered by AMS MathViewer

by Pace P. Nielsen PDF

Proc. Amer. Math. Soc. 135 (2007), 2689-2697 Request permission

Abstract:

Consider the ring of all $\kappa \times \kappa$ column finite matrices over a ring $R$. We prove that each such matrix is conjugate to a row and column finite matrix if and only if $R$ is right Noetherian and $\kappa$ is countable. We then demonstrate that one can perform this conjugation on countably many matrices simultaneously. Some applications and limitations are given.

References

P. Ara, K. C. O’Meara, and F. Perera, Gromov translation algebras over discrete trees are exchange rings, Trans. Amer. Math. Soc. 356 (2004), no. 5, 2067–2079. MR 2031053, DOI 10.1090/S0002-9947-03-03372-5
Victor Camillo, F. J. Costa-Cano, and J. J. Simón, Relating properties of a ring and its ring of row and column finite matrices, J. Algebra 244 (2001), no. 2, 435–449. MR 1859035, DOI 10.1006/jabr.2001.8901
K. R. Goodearl, P. Menal, and J. Moncasi, Free and residually Artinian regular rings, J. Algebra 156 (1993), no. 2, 407–432. MR 1216477, DOI 10.1006/jabr.1993.1082
J. Haefner, A. del Río, and J. J. Simón, Isomorphisms of row and column finite matrix rings, Proc. Amer. Math. Soc. 125 (1997), no. 6, 1651–1658. MR 1389521, DOI 10.1090/S0002-9939-97-03849-5
John Hannah and K. C. O’Meara, A new measure of growth for countable-dimensional algebras. I, Trans. Amer. Math. Soc. 347 (1995), no. 1, 111–136. MR 1282887, DOI 10.1090/S0002-9947-1995-1282887-9
W. K. Nicholson and K. Varadarajan, Countable linear transformations are clean, Proc. Amer. Math. Soc. 126 (1998), no. 1, 61–64. MR 1452816, DOI 10.1090/S0002-9939-98-04397-4
K. C. O’Meara, The exchange property for row and column-finite matrix rings, J. Algebra 268 (2003), no. 2, 744–749. MR 2009331, DOI 10.1016/S0021-8693(03)00266-7