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Published in The American Mathematical Monthly, Vol. 71, No. 10 (Dec., 1964), pp. 1112-1113 Copyright 1964 Mathematical Association of America Used by permission.


The following is another short proof of the fact that for a commutative ring with unit R, any finitely based R-module is "dimensional" in the sense that all of its bases have the same number of elements.

THEOREM. Let R be a commutative ring with unit. If M is a unitary R-module with a basis of n elements, then all bases of M contain exactly n elements.

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