3.1. Definitions.
In group theory, we started with a set \(G\) equipped with a bilinear operation \(\cdot : G \times G \to G\) which mapped \(G\) to itself. The operation was required to be associative, and there needed to be inverses and an identity element.
In ring theory, we went further to assume \(R\) was not only an abelian group, we placed the group operation with \(+: R\times R \to R\) and then defined a multiplication \(\cdot: R \times R \to R\) which is was associative and left- and right- distributive.
Finally, we reach module theory, which considers again an abelian group \(M\) with operation \(+:M \times M \to M\) but lets a ring \(R\) act on \(M\), whose addition \(+R\times R \to R\) agrees with the one which acts on \(M\) but whose multiplication \(\cdot: R \times M \to M\) acts on \(R\) and \(M\).
Note how abelian groups and rings are special cases of modules. This will be more clear once we introduce the axioms.
Let \(R\) be a ring with identity, and \(M\) an abelian group equipped with \(+:M \times M \to M\). Then \(M\) is an left \(R\)-module if we equip \(R \times M\) with multiplication \(\cdot : R \times M \to M\) and for all \(m \in M\) and \(a, b \in R\) \begin{enumerate} \item \(a(m_1 + m_2)= am_1 + am_2\) \item \((a + b)m = am + bm\) \item \((ab)m = a(bm)\) \item \(1_Rm = m\) where \(1_R\) is the identity of \(R\). \end{enumerate}
Alternatively, an abelian group \(M\) is a right \(R\)-module if we equip \(M \times R \to M\) with multiplication \(\cdot: M \times R \to M\) and for all \(m \in M\) and \(a, b \in R\) \begin{enumerate} \item \((m + n)a = ma + na\) \item \(m(a + b) = ma + mb\) \item \(m(ab) = (ma)b\) \item \(m1_R = m\) where \(1_R\) is the identity of \(R\). \end{enumerate}
Notice that we can think of these products as a group action, or sort of a "ring action" acting on \(M\). That is, an \(R\)-module \(M\) is just an abelian group that a ring \(R\) can act on. If you have an abelian group \(N\) that \(R\) simply cannot act on and satisfy the above axioms, then \(N\) is not an \(R\)-module. \ For convenience, we will develop the theory of \(R\)-modules by solely working with left \(R\)-modules, since all proofs and statements will be equivalent up to a swap of variables for right \(R\)-modules. \ Examples.\
- 1. \textcolor{NavyBlue}{Note that if \(R\) is commutative, then a left \(R\)-module coincides with a right \(R\)-module. To see this, let \(M\) be a left \(R\)-module. Then construct the right \(R\)-module by defining the multiplication as
Then we see that for all \(m \in M\), \(a,b \in R\), \begin{enumerate} * \((m_1 + m_2)\cdot a = a(m_1 + m_2) = am_1 + am_2 = m_1 \cdot a + m_2 \cdot a\) \checkmark * \(m(a + b)= (a + b)m = am + bm = m \cdot a + m \cdot b\) \checkmark * \(m\cdot (a b) = (ab)m = (ba)m = b(am) = b(m \cdot a) = (m \cdot a )\cdot b\) \checkmark * \(m \cdot 1_R = 1_Rm = m\). \checkmark \end{enumerate} Note that in part \((c)\) is where we used the fact that \(R\) is commutative. So whenever \(R\) is commutative, the existence of a left \(R\)-module automatically implies that existence of a right \(R\)-module, and vice versa. } * 2. Let \(R\) be a ring. Then if we substitute \(M =R\) in the above definition, and let the multiplication $ \cdot$ be the multiplication on \(R\) then \(R\) is a left and a right \(R\)-module. This is because \(R\) is an abelian group which is associative and left- and right-distributive. Hence, it satisfies all of the above axioms.
So keep in mind that a ring \(R\) is just a left- and right-\(R\) module that acts on \(R\).
Here's another example which shows that abelian groups are simply \(\ZZ\) modules.
Let \(G\) be an abelian group. Then \(G\) is a left and right \(\ZZ\)-module.
Let \(\ZZ\) act on \(G\) as follows. Define
and
Then with this definition of multiplication, it is easy to show that the axioms (a)-(d) are satisfied.
- 3. If \(R\) is a ring and \(I\) is a left (right) ideal of \(R\), then \(I\) is a left (right) \(R\)-module.
- 4. Let \(V\) be a vector space defined over a field \(F\). Then \(V\) is an \(F\)-module. (Now it is clear why there are a million axioms in the definition of a vector space!)
With \(R\)-modules introduced and understood, we can jump right into homomorphisms.
Let \(R\) be a ring and \(M\) and \(N\) be \(R\)-modules. We define \(f: M \to N\) to be an \(R\)-module homomorphism if
- 1. \(f(m_1 + m_2) = f(m_1) + f(m_2)\) for any \(m_1, m_2 \in M\)
- 2. \(f(am) = af(m)\) for all \(a \in R\) and \(m \in M\).
If \(f\) is a bijective \(R\)-module homomorphism, then we say that \(f\) is an isomorphism and that \(M \cong N\). Thus we see that \(R\)-module homomorphisms must not only be linear over the elements of \(M\), but they must also pull out scalar multiplication by elements of \(R\).
Recall earlier that we said a vector space \(V\) over a field \(F\) is an \(F\)-module. Now if \(W\) is another vector space and \(T: V \to W\) is a linear transformation, then we see that \(T\) is also an \(F\)-module homomorphism!
In the language of linear algebra, a linear transformation is usually defined as a function \(T: V \to W\) such that for any \(\bf{v}_1, \bf{v}_2, \bf{v} \in V\) and \(\alpha \in F\) we have that
- 1. \(T(\bf{v}_1 + \bf{v}_2) = T(\bf{v}_1) + T(\bf{v}_2)\)
- 2. $T(\alpha\bf{v}) = $ \(\alpha\)\(T(\bf{v})\).
As we will see, linear algebra is basically a special case of module theory.
Let \(R\) be a ring and \(M\) and \(N\) a pair of \(R\)-modules. Then \(\hom_R(M,N)\) is the set of all \(R\)-module homomorphisms from \(M\) to \(N\).
\textcolor{MidnightBlue}{It turns out we can turn \(\hom_R(M,N)\) into an abelian group, and under special circumstances it can actually be an \(R\)-module itself. It will be the case that \(\hom_R\) will actually be an important functor, but that is for later.} \ \indent To turn this into an abelian group, we define addition of the elements to be
for all \(f, g \in \hom_R(M, N)\). We let the identity be the zero map, and realize associativity and closedness are a given to conclude that this is in fact an abelian group. \
Suppose we want to make \(R\), our ring, act on \(\hom_R(M, N)\) in order for it to be an \(R\)-module. Then we define scalar multiplication to be \((af)(m) = a(f(m))\); a pretty reasonable definition for scalar multiplication.
\textcolor{Red}{This issue with this is that \(\hom_R(M, N)\) will not be closed under scalar multiplication of elements of \(R\) unless \(R\) is a commutative ring. }
We'll demonstrate this as follows. Let \(b \in R\) and \(f \in \hom_R(M, N)\). Then the second property of an \(R\)-module homomorphism tells us that \(f(bm) = bf(m)\) for all \(m \in M\). Now suppose we try to use our definition of scalar multiplication, and consider \(af\) where \(a \in R\). Then if we try to see if \(af\) will pass the second criterion for being an \(R\)-module homomorphism, we see that
That is, we see that \(af\) isn't an \(R\)-module homomorphism because \((af)(bm) \ne b(af)(m)\) (which is required for an \(R\)-module homomorphism); rather, \((af)(bm) = abf(m).\) Now if \(R\) is a commutative ring, then
so we can then say that \((af)(bm) = b(af)(m)\), in which case \(af\) passes the test for being an \(R\)-module homomorphism.
This proves the following propsition, which will be useful for reference for later.
Let \(M\) and \(N\) be \(R\)-modules. Then \(\hom_R(M, N)\) is an abelian group. Furthermore, it is an \(R\)-module if and only if \(R\) is a commutative ring. Next, we make the following definitions for completeness.
Let \(R\) be a ring and \(M\) and \(N\) be \(R\)-modules. If \(f: M \to N\) is an \(R\)-module homomorphism, then
- 1. The set \(\ker(f) = \{m \in M \mid f(m) = 0\}\) is the kernal of \(f\)
- 2. The set \(\im(f) = \{f(m) \mid m \in M\}\) is the image of \(f\).