lecture2

lecture2 - Lecture 2 (Aug 25, 2010) Definition 1....

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

This note was uploaded on 11/12/2010 for the course CSCI 271 taught by Professor Wilczynski during the Spring '08 term at USC.

Page1 / 2

lecture2 - Lecture 2 (Aug 25, 2010) Definition 1....

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online