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Algebra-II: Companion notes

Abhay Soman

Section 9.2 Examples

We show that every abelian group is a left \(\Z\)-module. Indeed, let \((A,+)\) be an abelian group. We define the map \(\Z\times A\to A\) by \((n,a)\mapsto\underbrace{a+\cdots+a}_{n-\text{times}}\) if \(n\in\N\) and \((-n,a)\mapsto\underbrace{(-a)+\cdots+(-a)}_{n-\text{times}}\) if \(n\in\N\text{.}\) It is left to the reader to verify that with this map \(A\) becomes \(\Z\)-module.
Conversely, any \(\Z\)-module is, by definition, an abelian group.
Let \(V\) be a finite-dimensional vector space over a field \(F\text{.}\) Then \(V\) is also a left \(F\)-module. In other words, left modules over a field are vector spaces.
Let \(V\) be an \(F\)-vector space. We define the map:
\begin{equation*} \End_F(V)\times V\to V;\quad (T,v)\mapsto T(v). \end{equation*}
Verify that with this map \(V\) is a left \(\End_F(V)\)-module.
Let \(V\) be a vector space over a field \(F\text{,}\) and fix an \(F\)-linear map \(T\in\End_F(V)\text{.}\) Recall that \(T^r\) is the linear map given by \(\underbrace{T\circ\cdots\circ T}_{r-\text{times}}\text{.}\) We define a map
\begin{equation*} F[X]\times V\to V \end{equation*}
by
\begin{equation*} (a_0+a_1X+\cdots+a_nX^n,v)\mapsto a_0v+a_1T(v)+a_2T^2(v)+\cdots+a_nT^n(v). \end{equation*}
It is left to the reader to verify that this makes \(V\) a left \(F[X]\)-module.
Let \(R\) be a ring. The ring multiplication maps
\begin{equation*} R\times R\to R\quad\text{given by}\quad (r,s)\mapsto rs \end{equation*}
defines a left and a right \(R\)-module structure on \(R\text{.}\)
Let \((A,+)\) be an abelian group. The group of all group endomorphisms of \(A\text{,}\) \(\End_{\rm Grps}(A)\) is a ring. Assume that there is a ring homomorphism \(\varphi\colon R\to\End_{\rm Grps}(A)\text{.}\) We make \(A\) as an \(R\)-mod using \(\varphi\text{.}\) Indeed, consider the following map:
\begin{equation*} R\times A\to A\quad\text{defined by}\quad (r,a)\mapsto\varphi(r)(a). \end{equation*}
This map makes \(A\) into an \(R\)-mod. The verification is left to the reader.
Conversely, suppose that \(A\) is a left \(R\)-module. For \(r\in R\) define a group homomorphism \(f_r\colon A\to A\) by \(f_r(a)=r\cdot a\text{.}\) We get a ring homomorphism
\begin{equation*} f\colon R\to\End_{\rm Grps}(A)\quad\text{given by}\quad r\mapsto f_r. \end{equation*}
All verifications are left to the reader.
Thus, there is a left \(R\)-module structure on an abelian group \(A\) if and only if there is a ring homomorphism \(R\to\End_{\rm Grps}(A)\text{.}\)
Let \(\varphi\colon R\to S\) be a ring homomorphism, and let \(N\) be a left \(S\)-module. We show that \(N\) can be considered as a left \(R\)-module. Indeed, define
\begin{equation*} R\times N\to N\quad\text{by}\quad (r,n)\mapsto\varphi(r)\cdot n\text{.} \end{equation*}
Verify that this makes \(N\) a left \(R\)-module.
If we consider the identity homomorphism on \(R\) then we get Example 9.2.5.
Let \(R\) be a ring and \(R^{\rm op}\) be its opposite ring (see Example 1.2.9). If \(M\) is a left \(R\)-module (resp., a right \(R\)-module) then \(M\) is a right \(R^{\rm op}\)-module (resp., a left \(R^{\rm op}\)-module). Indeed, given a left \(R\)-module \(M\) we define \(M\times R^{\rm op}\to M\) by
\begin{equation*} (m,r^{\rm op})\mapsto r\cdot m. \end{equation*}
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