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.

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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 $R$-module]

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

1. a binary operation $+$ on $M$ under which $(M, +)$ is an abelian group, and
2. an action of $R$ on $M$ (a map $R \times M \rightarrow M$) denoted by $rm$, for all $r \in R$ and for all $m \in M$ which satisfies
   • $(r+s)m = rm + sm$ for all $r, s \in R, m \in M$;
   • $(rs)m = r(sm)$ for all $r, s \in R, m \in M$;
   • $r(m+n) = rm + rn$ for all $r \in R$, $m, n \in M$;
3. if the ring $R$ has a $1$ we impose the additional axiom
   • $1m = m$ for all $m \in M$;

Definition. [$R$-submodule]

Let $R$ be a ring and let $M$ be an $R$-module. An $R$-submodule of $M$ is a subgroup $N$ of $M$ which is colsed under the action of ring elemtns. That is, $rn \in N$ for all $r \in R$, $n \in N$.