Vector spaces and ring modules

Overview and basic definitions

Definition: Vector space

Given a field $F$ (the scalars), a $F$-vector space, or simply vector space, is a triple $(V,+,\cdot )$ where $(V,+)$ is an abelian group (the vectors) and $\cdot :F\times V\to V$ is a map $(t,v)\mapsto t\cdot v$ (scalar multiplication) such that for all $s,t\in F$ and $v,w\in V$, we have the following:

• (i) Associativity of multiplication: $s(tv)=(st)v$;
• (ii) Distributivity of multiplication over scalar addition: $(s+t)v=sv+tv$;
• (iii) Distributivity of multiplication over vector addition: $s(v+w)=sv+sw$;
• (iv) Identity: $1\cdot v=v$ for the unit $1\in F$.

Definition: Vector subspace

Given a field $F$ and an $F$-vector space $V$, an $F$-vector subspace is a subset $W\subseteq V$ with the following properties:

• (i) Subgroup under $+$: $W$ is closed under addition and contains the additive identity $0\in V$ (hence non-empty);
• (ii) Closure under multiplication: For all $w\in W$ and all $t\in F$, we have $tw\in W$ as well.

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Analogue in general rings

Definition: Module

Given a commutative ring with unity $R$, a $R$-module is a triple $(M,+,\cdot )$, where $(M,+)$ is an abelian group and $\cdot :R\times M\to M$ is a map $(r,m)\mapsto r\cdot m$ such that for all $r,s\in R$ and $m\in M$, the analogue of each condition for vector spaces holds.

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Examples

General $F$-vector spaces

If $F$ is a field, then the following are $F$-vector spaces:

• The $n$-fold Cartesian product $F^n=F\times \cdots \times F$with component-wise addition and scalar multiplication defined by$(a_1,\dots ,a_n)+(b_1,\dots ,b_n)=(a_1+b_1,\dots ,a_n+b_n),$ $t(a_1,\dots ,a_n)=(t{a}_1,\dots ,t{a}_n),$ respectively.
• The group of functions $F^X$ from $X\to F$.
• The polynomial ring $F[x]$.
• Any ring $R$ containing $F\subseteq R$ as a subring.

General $R$-modules

If $R$ is a commutative ring with unity, then the following are $R$-modules:

• Both an ideal $I\subseteq R$ and the corresponding quotient ring $R/I$.