Modules
Key Definitions
Definition.
Let $R$ be a commutative ring with unity. An abelian group $M$ is called a (left) $R$–module if there exists a scalar multiplication map
satisfying:
- $r \cdot (m_1 + m_2) = r \cdot m_1 + r \cdot m_2$
- $(r+s) \cdot m = r \cdot m + s \cdot m$
- $(rs) \cdot m = r \cdot (s \cdot m)$
- $1 \cdot m = m$
An $R$–module structure on $M$ is equivalent to a unital ring homomorphism
\[R \to \operatorname{End}(M).\]If $R = k$ is a field, then an $R$–module is precisely a vector space over $k$.
Definition.
Let ${M_i}_{i \in I}$ be $R$–submodules of $M$. The sum of the $M_i$ is defined as
We say the sum is an (internal) direct sum if
\[M_i \cap \sum_{j \in I, \ j \neq i} M_j = \{0\} \quad \text{for all } i \in I.\]In this case, we write
\[\bigoplus_{i \in I} M_i.\]The internal direct sum is isomorphic to the external direct sum
\[\bigoplus_{i \in I} M_i = \{ (m_i)_{i \in I} : m_i \in M_i, \ \text{finitely many } m_i \neq 0 \}.\]Thus, we do not distinguish between internal and external direct sums in notation.
Definition.
For a family of $R$–modules ${M_i}_{i \in I}$, the direct product is defined as
Definition.
Let $M$ be an $R$–module and $X \subset M$. We say $X$ generates $M$ if
Equivalently, the natural $R$–linear map
\[\phi : R^{|X|} \to M, \qquad (c_x)_{x \in X} \mapsto \sum_{x \in X} c_x x\]is surjective.
- $X$ freely generates $M$ if $M=\sum_{x \in X} R x$ is a direct sum. In this case, we say that $M$ is a free $R$-module and $M \cong \bigoplus_{x\in X}Rx\cong R^{|X|}$ as $R$–modules. We call $X$ as a basis of $M$.
- $M$ is finitely generated if there exists a finite generating set $X$.
Definition.
Let $M$ be an $R$–module. A subset $X \subset M$ is linearly independent if $\sum_{x \in X} R x$ is a direct sum.
The rank of $M$ is the cardinality of the largest linearly independent subset of $M$.