Rings and Ideals - Core Definitions

Rings extend the structure of abelian groups by adding a second operation (multiplication), providing the algebraic framework for arithmetic and polynomial algebra.

DefinitionRing

A ring is a set $R$ together with two binary operations $+$ (addition) and $\cdot$ (multiplication) satisfying:

$(R, +)$ is an abelian group with identity $0$
Multiplication is associative: $(ab)c = a(bc)$
Distributive laws: $a(b+c) = ab + ac$ and $(a+b)c = ac + bc$

If multiplication is commutative, $R$ is a commutative ring. If there exists $1 \in R$ with $1a = a1 = a$ for all $a$ , then $R$ is a ring with identity (or unital ring).

ExampleClassical Rings

Integers: $\mathbb{Z}$ with usual addition and multiplication
Polynomials: $\mathbb{Z}[x], \mathbb{Q}[x], \mathbb{R}[x]$ (polynomials with integer, rational, real coefficients)
Matrix rings: $M_n(R)$ ( $n \times n$ matrices over a ring $R$ )
Function rings: $C([0,1])$ (continuous real functions on $[0,1]$ )
Modular integers: $\mathbb{Z}/n\mathbb{Z}$ (integers modulo $n$ )

DefinitionIdeal

A subset $I \subseteq R$ of a ring $R$ is a left ideal if:

$(I, +)$ is a subgroup of $(R, +)$
For all $r \in R$ and $a \in I$ , we have $ra \in I$

Similarly, $I$ is a right ideal if $ar \in I$ for all $r \in R, a \in I$ . An ideal that is both left and right is a two-sided ideal (or simply an ideal).

For commutative rings, left, right, and two-sided ideals coincide.

Ideals generalize the notion of "multiples" from $\mathbb{Z}$ . Just as $n\mathbb{Z} = \{nk : k \in \mathbb{Z}\}$ consists of all multiples of $n$ , an ideal consists of elements that can be "multiplied into" from the outside.

ExamplePrincipal Ideals

For $a \in R$ , the principal ideal generated by $a$ is: $(a) = \{ra : r \in R\} = Ra$

In $\mathbb{Z}$ , every ideal has the form $(n) = n\mathbb{Z}$ for some $n \geq 0$ . In polynomial rings $F[x]$ over a field $F$ , every ideal is also principal.

Remark

Ideals play the role for rings that normal subgroups play for groups: they are precisely the kernels of ring homomorphisms and allow the construction of quotient rings. The study of ideal structure reveals deep properties of the ring.

Understanding rings requires recognizing both their additive structure (an abelian group) and their multiplicative structure, along with how these interact via the distributive laws.