Modules

We now define \(A\)-modules for an arbitrary ring \(A\) and investigate their properties. More advanced content on \(A\)-modules can be found in the Multilinear Algebra category.

Definition 1 Fix a monoid object \((A,\cdot, 1)\) in the symmetric monoidal category \((\Ab,\otimes, \mathbb{Z})\). An object \(M\in\Ab\) equipped with a left \(A\)-action is called a left \(A\)-module, and one equipped with a right \(A\)-action is called a right \(A\)-module.

Since a monoid object in the monoidal category \((\Ab,\otimes, \mathbb{Z})\) is precisely a ring \(A\), rewriting the above definition for this case means that \(M\) is a left \(A\)-module if \(M\) is an abelian group under addition and is equipped with an action \(\cdot:A\times M \rightarrow M\) of \(A\) satisfying the following conditions:

1. For any \(\alpha\in A\) and \(x,y\in M\), we have \(\alpha\cdot(x+y)=\alpha\cdot x+\alpha\cdot y\).
2. For any \(\alpha,\beta\in A\) and \(x\in M\), we have \((\alpha+\beta)\cdot x=\alpha\cdot x+\beta\cdot x\).
3. For any \(\alpha,\beta\in A\) and \(x\in M\), we have \((\alpha\beta)\cdot x=\alpha\cdot(\beta\cdot x)\).
4. For any \(x\in M\), we have \(1\cdot x=x\).

Here, instead of viewing \(\cdot\) as a homomorphism from \(A\otimes M\) to \(M\), we have explicitly stated the bilinearity condition in conditions 1 and 2.

Definition 2 For an \(A\)-module \(M\) and a family \((x_i)_{i\in I}\) of its elements, a linear combination of this family is an element of the form

\[\sum_{i\in I} \alpha_i x_i,\qquad\text{$(\alpha_i)$ finitely supported}\]

If there exists a family \((x_i)_{i\in I}\) of elements of \(M\) such that every element of \(M\) can be expressed as a linear combination of elements from this family, we say that \((x_i)_{i\in I}\) generates \(M\). If this family \((x_i)_{i\in I}\) can be chosen to be finite, then \(M\) is called a finitely generated \(A\)-module.

Submodules and Quotient Modules

Definition 3 For any \(A\)-module \(M\) and its subset \(N\), if \(N\) itself carries an \(A\)-module structure, it is called a submodule.

On the other hand, for an \(A\)-module \(M\) and any submodule \(N\) of \(M\), the quotient group \(M/N\) is well-defined. Considering the action of \(A\) on this quotient, since \(N\) is closed under the \(A\)-action, there is no issue in giving \(M/N\) an \(A\)-module structure.

Definition 4 For any \(A\)-module \(M\) and its submodule \(N\), the quotient \(M/N\) is called the quotient module.

Example 5 The multiplication map \(\mu:A\otimes A \rightarrow A\) of a ring \(A\) satisfies exactly all the properties that an action should satisfy. Therefore, any ring is always a module over itself. If we regard \(A\) as a left \(A\)-module, then the submodules of \(A\) are precisely the left ideals of \(A\), and similarly, if we regard \(A\) as a right \(A\)-module, then the submodules of \(A\) are precisely the right ideals of \(A\).

On the other hand, reconsidering why we only considered two-sided ideals when defining quotient rings, we can see that for a left ideal \(\mathfrak{a}\), the quotient module \(A/\mathfrak{a}\) (even if it does not carry a ring structure) is still a left \(A\)-module, and the same holds for right ideals.

Linear Maps

Definition 6 For two \(A\)-modules \(M\) and \(N\), an \(A\)-linear map from \(M\) to \(N\) is a function \(u\) satisfying the two conditions

\[u(x+y)=u(x)+u(y),\qquad u(\alpha x)=\alpha u(x)\]

for all \(x,y\in M\) and \(\alpha\in A\). When there is no room for confusion, this may simply be called a linear map.

Proposition 7 The composition of linear maps is a linear map. Also, a bijective linear map is always an isomorphism.

Proof

Clear.

The category of left \(A\)-modules and \(A\)-linear maps is denoted by \(\lMod{A}\). The category of right \(A\)-modules and \(A\)-linear maps is denoted by \(\rMod{A}\). Also, \(\Hom_\lMod{A}(M,N)\) is sometimes simply written as \(\Hom_A(M,N)\). These have full subcategories \(\lmod{A}\) and \(\rmod{A}\) consisting of finitely generated \(A\)-modules. The zero object of these four categories is \(\{0\}\).

On the other hand, one special property of \(\lMod{A}\) is that \(\Hom_{\lMod{A}}(M,N)\) carries an abelian group structure. Moreover, the following holds.

Proposition 8 For any \(\Hom_{\lMod{A}}(M,N)\), it is an abelian group. Moreover, for any \(A\)-linear map \(u:M \rightarrow M'\),

\[\Hom_{\lMod{A}}(u, N):\Hom_{\lMod{A}}(M',N)\rightarrow \Hom_{\lMod{A}}(M,N)\]

is a homomorphism between abelian groups.

Proof

The sum \(v+w\in\Hom_{\lMod{A}}(M,N)\) of any two elements \(v,w\in\Hom_{\lMod{A}}(M,N)\) is defined by the formula

\[(v+w)(x)=v(x)+w(x)\qquad\text{for all $x\in M$}\]

We need to verify that this is indeed an \(A\)-linear map, but this is clear.

Now, that \(\Hom_{\lMod{A}}(u,N)\) is a homomorphism between abelian groups follows from the equation

\[\left(\Hom_{\lMod{A}}(u, N)(v+w)\right)(x)=(v+w)(u(x))=v(u(x))+w(u(x))=\left(\Hom_{\lMod{A}}(u,N)(v)\right)(x)+\left(\Hom_{\lMod{A}}(u,N)(w)\right)(x)\]

A similar theorem holds for \(\Hom_{\lMod{A}}(M, f)\) and for right \(A\)-modules. If \(A\) is a commutative ring, then for any \(u:M \rightarrow N\), the operation \(\cdot\) defined by

\[(\alpha\cdot u)(x):=\alpha\cdot u(x)\qquad\text{for all $x\in M$}\]

gives \(\Hom_{\lMod{A}}(M,N)\) an \(A\)-module structure. However, if \(A\) is not commutative, then for any \(\beta\in A\),

\[(\alpha\cdot u)(\beta x)=\alpha\cdot u(\beta x)=\alpha\beta u(x)\]

and there is no natural way to convert this to \(\beta(\alpha u(x))=\beta\cdot(\alpha\cdot u)(x)\), so \(\Hom_{\lMod{A}}(M,N)\) does not generally carry an \(A\)-module structure. On the other hand, if \(A\) is commutative, we can even show that the abelian group homomorphism from Proposition 8 is an \(A\)-linear map.

Definition 9 For an \(A\)-linear map \(u:M \rightarrow N\), the kernel and image of \(u\) are defined respectively as

\[\ker u=\{x\in M\mid u(x)=0\},\qquad \im u=\{u(x)\in N\mid x\in M\}\]

The following is the isomorphism theorem we have always used, and since its proof is the same as before, we will not write it separately.

Theorem 10 Let an \(A\)-linear map \(u:M \rightarrow N\) be given.

1. \(\ker u\) is a submodule of \(M\), and \(x+\ker u \mapsto u(x)\) defines a well-defined isomorphism \(M/\ker u \rightarrow \im u\).
2. For two submodules \(M',M''\) of \(M\), the sum \(M'+M''\) and intersection \(M'\cap M''\) are submodules of \(M\), and an isomorphism \((M'+M'')/M''\cong M'/(M'\cap M'')\) holds.
3. If two submodules \(M',M''\) of \(M\) satisfy \(M''\subseteq M'\), then \(M'/M''\) is a submodule of \(M/M''\) and \((M/M'')/(M'/M'')\cong M/M'\) holds.
4. For a submodule \(M'\) of \(M\), there is an inclusion-preserving bijection between the set of submodules of \(M/M'\) and the set of submodules of \(M\) containing \(M'\).

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References

[Bou] Bourbaki, N. Algebra I. Elements of Mathematics. Springer. 1998.

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