Rings, Ideals, and Modules - Core Definitions

Commutative algebra studies commutative rings and their modules, providing the algebraic foundation for algebraic geometry and number theory.

DefinitionCommutative Ring

A commutative ring is a set $R$ equipped with two binary operations, addition $+$ and multiplication $\cdot$ , satisfying:

$(R, +)$ is an abelian group with identity $0$
Multiplication is associative and commutative
There exists a multiplicative identity $1 \neq 0$
Distributivity: $a(b + c) = ab + ac$ for all $a, b, c \in R$

Unless stated otherwise, all rings in this text are commutative with identity.

ExampleFundamental Examples

Integers $\mathbb{Z}$ : The prototypical commutative ring
Fields $\mathbb{Q}, \mathbb{R}, \mathbb{C}$ : Rings where every non-zero element is invertible
Polynomial rings $k[x_1, \ldots, x_n]$ : Central objects in algebraic geometry
Ring of integers $\mathcal{O}_K$ in a number field $K$ : Foundation of algebraic number theory

DefinitionIdeal

A subset $I \subseteq R$ is an ideal if:

$I$ is an additive subgroup of $R$
For all $r \in R$ and $a \in I$ , we have $ra \in I$

An ideal $I$ is proper if $I \neq R$ . A proper ideal $\mathfrak{p}$ is prime if $ab \in \mathfrak{p}$ implies $a \in \mathfrak{p}$ or $b \in \mathfrak{p}$ . An ideal $\mathfrak{m}$ is maximal if it is maximal among proper ideals.

DefinitionModule

An $R$ -module $M$ is an abelian group $(M, +)$ equipped with scalar multiplication $R \times M \to M$ satisfying:

$r(m + n) = rm + rn$
$(r + s)m = rm + sm$
$(rs)m = r(sm)$
$1 \cdot m = m$

for all $r, s \in R$ and $m, n \in M$ . Modules generalize vector spaces by replacing fields with rings.

Remark

The quotient $R/I$ by an ideal $I$ forms a ring. Maximal ideals correspond to fields via $R/\mathfrak{m}$ , while prime ideals correspond to integral domains via $R/\mathfrak{p}$ . This connection between ideals and quotient structures is fundamental throughout commutative algebra.

The interplay between rings, ideals, and modules forms the backbone of commutative algebra, with ideals encoding geometric and arithmetic properties of rings, and modules providing a flexible framework for studying linear algebra over rings.