Unformatted text preview: n It is possible for a free module over an non-commutative ring with identity to have two bases of different cardinalities. See Exercise 8.4.4 . Deﬁnition 8.4.2. Let R be a commutative ring with identity element. The rank of a ﬁnitely generated free R –module is the cardinality of any basis. Remark 8.4.3. The zero module over R is free of rank zero. The empty set is a basis. This is not just a convention; it follows from the deﬁnitions. For the rest of this section, R denotes a principal ideal domain. Lemma 8.4.4. Let F be a free module of ﬁnite rank n over a principal idea domain R . Any submodule of F has a generating set with no more than n elements...
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- Fall '08
- Algebra, Ring, single matrix equation, Wm B, AB D En, n–by–n identity matrix