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Unformatted text preview: Lecture 2 (Aug 25, 2010) Definition 1. (Submodules, quotient modules, canonical mappings). A submodule F of an A-module E, is an additive subgroup which is stable under scalar multipli- cation. If F E is a sub-module, the quotient module E / F is defined as the quo- tient group, with scalar multiplication a ( x + F ) = ax + F . The linear map E E / F , a a + F , is called the canonical mapping (surjection). Examples: 1. is always a submodule. 2. The submodules of A are simply the left ideals of A . 3. If I is a left ideal of A and x E , then I x E is a sub- module. 4. In a commutative group G , considered as a Z-module, every subgroup is also a sub-module. (In general an additive sub- group of E is not necessarily a sub-module). Let u : E F be a linear mapping. Then the image (resp. the preimage) of a sub- module of E (resp. of F ) is a submodule of F (resp. of E ). In particular, the image u ( E ) (denoted Im ( u ) ) and u- 1 (0) (denoted Ker ( u ) ) are sub-modules. We) are sub-modules....
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This note was uploaded on 11/12/2010 for the course CSCI 271 taught by Professor Wilczynski during the Spring '08 term at USC.
- Spring '08