Section 9.1 Definition
We generalize the concept of a vector space over a field.
Definition 9.1.1. (Left Module).
Let \(R\) be a ring. An abelian group \((M,+)\) is said to be a left \(R\)-module if there is a map ‘scalar multiplication’
\begin{equation*}
R\times M\to M\quad\text{denoted by}\quad(r,m)\mapsto r\cdot m
\end{equation*}
satisfying the following conditions:
For any
\(r\in R\) and any
\(m,n\in M\)
\begin{equation*}
r\cdot(m+n)=r\cdot m+r\cdot n
\end{equation*}
For any
\(r,s\in R\) and any
\(m\in M\)
\begin{equation*}
(r+s)\cdot m=r\cdot m+s\cdot m
\end{equation*}
For any
\(r,s\in R\) and any
\(m\in M\)
\begin{equation*}
(rs)\cdot m=r\cdot(s\cdot m)
\end{equation*}
For any
\(m\in M\)
\begin{equation*}
1_R\cdot m=m
\end{equation*}
In a similar way we define a right module.
Definition 9.1.3. (Right Module).
Let \(R\) be a ring. An abelian group \((M,+)\) is said to be a right \(R\)-module if there is a map ‘scalar multiplication’
\begin{equation*}
M\times R\to M\quad\text{denoted by}\quad(m,r)\mapsto m\cdot r
\end{equation*}
satisfying the following conditions:
For any
\(r\in R\) and any
\(m,n\in M\)
\begin{equation*}
(m+n)\cdot r= m\cdot r+ n\cdot r
\end{equation*}
For any
\(r,s\in R\) and any
\(m\in M\)
\begin{equation*}
m\cdot (r+s)=m\cdot r+ m\cdot s
\end{equation*}
For any
\(r,s\in R\) and any
\(m\in M\)
\begin{equation*}
m\cdot (rs)=(m\cdot r)\cdot s
\end{equation*}
For any
\(m\in M\)
\begin{equation*}
m\cdot 1=m
\end{equation*}
Following properties follows immediately from the definition of \(R\)-module.
Lemma 9.1.5.
Let \(M\) be a left \(R\)-module. For any \(r\in R\) and any \(m\in M\) we have the following.
\(\displaystyle r\cdot 0_M=0_M\)
\(\displaystyle 0_R\cdot m=0_M\)
\(\displaystyle r\cdot(-m)=-(r\cdot m)\)
For
\(m_1,\ldots,m_k\in M\) and any
\(r\in R\) we have
\begin{equation*}
r\cdot\left(m_1+\cdots +m_k\right)=r\cdot m_1+r\cdot m_2+\cdots+r\cdot m_k.
\end{equation*}
For
\(m_1,\ldots,m_\ell,n_1,\ldots,n_\ell\in M\) we have
\begin{equation*}
\sum_{i=1}^{\ell}(m_i+n_i)=\sum_{i=1}^{\ell}m_i+\sum_{i=1}^{\ell}n_i\text{.}
\end{equation*}
