Modules Over Commutative Rings

Abstract

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 i s a unitary R-module with a basis of n elements, then all bases of M contain exactly n elements.Proof.(The method is that of [I], p. 115.)Let {ail (i = 1, . . ., n ) be a basis for M. I t is easy to see that M cannot have an infinite basis.(See 121, p. 241-2.Applied to modules, the method shows that for a module with an infinite basis all bases have the same cardinality.)Thus let (flj] (j= 1, ., ., m ) be another