Introduction to Module Theory

I took Abstract Algebra I/II in UW-Madison(MATH541) with professor Chenxi Wu and Tonghai Yang. This article is inspired by prof. Wu's lecture note, and many formats, content, and structure of this blog are influenced by and similar to it.

RECAP

Please read this introduction with a complete understanding of ring and concepts including Euclidean / Principal Ideal / Unique Factorization Domains.

Basic Definition and Examples

Definition. [Left -module]

Let be a ring. A left -module or a left module over is a set together with

1. a binary operation on under which is an abelian group, and
2. an action of on (a map ) denoted by , for all and for all which satisfies
   • for all ;
   • for all ;
   • for all , ;
3. if the ring has a we impose the additional axiom
   • for all ;

Definition. [-submodule]

Let be a ring and let be an -module. An -submodule of is a subgroup of which is colsed under the action of ring elemtns. That is, for all , .