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The Quarterly Journal of Mathematics 2004 55(4):491-498; doi:10.1093/qmath/hah007
© 2004 by Oxford University Press
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Uniform dimension of modules

Roy L. McCasland1 * and Patrick F. Smith2 §

1 School of Informatics, University of Edinburgh, Edinburgh EH8 9LE, 2 Department of Mathematics, University of Glasgow, Glasgow G12 8QW  *

Let M be a module which has finite uniform dimension and let Ki(1 ≤ i ≤ n) be a finite collection of submodules of M such that 0 = K1 {cap}···{cap} Kn. Then the uniform dimension u(M) of M is the sum of the uniform dimensions of the factor modules M/Ki(1 ≤ i ≤ n) if and only if Ki is a complement of K1 {cap}···{cap} Ki–1 {cap} Ki+1 {cap}···{cap} Kn in M for each 1 ≤ i ≤ n. In case Ki is Pi-prime for some prime ideal Pi for each 1 ≤ i ≤ n, the prime ideals Pi (1 ≤ i ≤ n) are distinct and 0 != K1 {cap}···{cap} Ki–1 {cap} Ki+1 {cap}···{cap} Kn for each 1 ≤ i ≤ n, then it is shown that u(M) = {sum}i=1n u(Li/(Li {cap} Ki)) for certain submodules Li (1 ≤ i ≤ n) of M.

Received 20 April 2003. Revised 19 November 2003.

* E-mail: rmccasla{at}inf.ed.ac.uk

§ E-mail: pfs{at}maths.gla.ac.uk


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