This preview shows page 1. Sign up to view the full content.
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 . Denition 8.4.2. Let R be a commutative ring with identity element. The rank of a nitely 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 denitions. For the rest of this section, R denotes a principal ideal domain. Lemma 8.4.4. Let F be a free module of nite rank n over a principal idea domain R . Any submodule of F has a generating set with no more than n elements...
View Full Document
This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
- Fall '08