Commutative algebra studies the structure of commutative rings and their modules, laying the algebraic foundation for algebraic geometry and number theory. This course builds systematically from basic definitions through to dimension theory, equipping you with both foundational concepts and sophisticated tools for understanding polynomial rings, integral extensions, and the geometry hidden within algebraic structures. The chapters progress from concrete ring-theoretic properties toward increasingly abstract structural results, with a clear arc from local phenomena (localization, primary decomposition) to global invariants (dimension, integral closure).

The course begins with essentials: rings, ideals, and modules as the central objects of study, then introduces finiteness conditions through chain conditions that force rings and modules to behave well. Tensor products appear early as a fundamental operation, while localization emerges as a technique to isolate local behavior — a theme that recurs throughout. Nakayama's lemma and integral extensions anchor the middle chapters, providing tools to relate rings to their integral closures and to understand when a ring extension is finite. The Cohen-Seidenberg theorems capture how dimension and prime ideals behave under integral extensions, while primary decompositions generalize unique factorization beyond factorial rings.

Later chapters shift focus to homological and invariant structures: direct and inverse limits formalize taking colimits and their duals, completions produce new rings from old ones, and filtrations and gradings organize rings by degree or other decorations. Dimension theory synthesizes much of what comes before, quantifying the "size" of a ring through the lengths of chains of prime ideals.

# 1. Rings and Modules

This chapter sets up the two central objects of commutative algebra: modules over a commutative ring, and the finiteness conditions — the noetherian and artinian properties — that make them tractable. Modules generalise both vector spaces and abelian groups within a single framework, and the noetherian condition is the precise algebraic substitute for compactness. The chapter concludes with Hilbert's basis theorem, which is the engine behind the finiteness properties of polynomial rings and algebraic varieties.

Throughout, a **ring** means a commutative ring $R$ with $1 \in R$, unless stated otherwise. The one non-commutative exception that appears is $\operatorname{End}(M)$: for an abelian group $(M, +)$, the set $\operatorname{End}(M) = \{f: M \to M \mid f \text{ is a group homomorphism}\}$ forms a ring under composition as multiplication.

## Modules and the Structure Homomorphism

The idea behind an $R$-module is to have a set on which $R$ acts by scalar multiplication, generalising the notion of a vector space over a field. The key insight is to package this action as a single ring homomorphism.

[definition: R-Module] Let $R$ be a commutative ring with $1$. An **$R$-module** is an abelian group $M$ together with a fixed ring homomorphism $\rho: R \to \operatorname{End}(M)$, called the **structure homomorphism**. The scalar multiplication is defined by \begin{align*} r \cdot m := \rho(r)(m) \end{align*} for $r \in R$ and $m \in M$. [/definition]

The structure homomorphism viewpoint is conceptually clean: it packages all of the scalar multiplication axioms into the single requirement that $\rho$ be a ring homomorphism. Unpacking this, one obtains the familiar identities. Since $\rho(r) \in \operatorname{End}(M)$ is a group homomorphism, we have $r(m + m') = rm + rm'$. Since $\rho$ is a ring homomorphism, $\rho(r_1 + r_2) = \rho(r_1) + \rho(r_2)$ in $\operatorname{End}(M)$, giving $(r_1 + r_2)m = r_1 m + r_2 m$. Multiplicativity of $\rho$ gives $(r_1 r_2)m = r_1(r_2 m)$, and $\rho(1_R) = \operatorname{id}_M$ gives $1 \cdot m = m$.

[example: Standard Examples of Modules] The following illustrate the range of the module concept. **Vector spaces.** A $k$-module is exactly a $k$-vector space, for $k$ a field. The structure homomorphism is the natural scalar multiplication. **Abelian groups as $\mathbb{Z}$-modules.** Every abelian group $M$ is a $\mathbb{Z}$-module in a unique way. A ring homomorphism $\rho: \mathbb{Z} \to \operatorname{End}(M)$ is completely determined by $\rho(1_{\mathbb{Z}}) = \operatorname{id}_M$, so there is no choice. Conversely, any $\mathbb{Z}$-module is an abelian group. Thus: abelian group $\iff$ $\mathbb{Z}$-module. **The ring itself.** Every ring $R$ is an $R$-module via the structure homomorphism \begin{align*} R &\to \operatorname{End}(R) \\ r_0 &\mapsto (r \mapsto r_0 r). \end{align*} **Direct sums and products.** The direct sum $R^{\oplus \mathbb{N}}$ (elements with only finitely many non-zero coordinates) and the direct product $R^{\mathbb{N}}$ (all $\mathbb{N}$-indexed tuples) are both $R$-modules, with componentwise scalar multiplication. **Ideals.** An ideal of the ring $R$ is exactly the same thing as an $R$-submodule of $R$ (as an $R$-module). This unifies the theory of ideals with the theory of submodules. [/example]

[definition: Submodule] An **$R$-submodule** of an $R$-module $M$ is an additive subgroup $N \subseteq M$ such that $rN \subseteq N$ for all $r \in R$. [/definition]

For a subset $S$ of an $R$-module $M$, the $R$-submodule **generated by $S$** is the intersection of all $R$-submodules of $M$ containing $S$. Equivalently, it is the set of all $R$-linear combinations

\begin{align*} \left\{ \sum_{i=1}^{n} r_i x_i : n \geq 0,\, r_i \in R,\, x_i \in S \right\}. \end{align*}

Note that each sum is finite. A module is **finitely generated** if it is generated by a finite set $S = \{x_1, \dots, x_k\}$.

## Quotient Modules and Ideals

When can we collapse part of a module to zero without losing the $R$-module structure? The obstacle is that a bare subgroup quotient $M/N$ need not respect scalar multiplication: if $N$ is not closed under the $R$-action, the coset $r \cdot (m + N)$ is not well-defined. Requiring $rN \subseteq N$ for all $r \in R$ is precisely what fixes this — it is the definition of a submodule. For the special case where $M = R$ and $N = I$ is an ideal, we recover the familiar ring-quotient construction.

Given an ideal $I \trianglelefteq R$, the quotient $R/I$ inherits a ring structure. As an abelian group, $R/I$ is the usual group quotient. The multiplication is $(r_1 + I)(r_2 + I) = r_1 r_2 + I$. One should check this is well-defined: the product set $\{(r_1 + x_1)(r_2 + x_2) : x_1, x_2 \in I\}$ is contained in $r_1 r_2 + I$ (since $r_1 x_2 + r_2 x_1 + x_1 x_2 \in I$), so the product of two cosets lands in a single coset. Note that this containment may be proper, unlike in the group case where $g_1 N \cdot g_2 N = g_1 g_2 N$ exactly.

[remark: Characterisation of Ideals via Kernels] A subset $I \subseteq R$ is an ideal if and only if there exists a ring $S$ and a ring homomorphism $\varphi: R \to S$ whose kernel is $I$. The forward direction is the first isomorphism theorem: $\ker \varphi \trianglelefteq R$. The backward direction: take $S = R/I$ and $\varphi$ the quotient map. [/remark]

## Free Modules and Homomorphisms from $R^{\oplus n}$