Ideal

An ideal of a ring $R$ is a subset $I \subseteq R$ that is an additive subgroup of $(R, +)$ and absorbs multiplication: for every $r \in R$ and $a \in I$, both $ra$ and $ar$ belong to $I$.

Ideals play the same role in ring theory that normal subgroups play in group theory: they are the substructures you can “divide out” to form a quotient. The quotient ring $R/I$ consists of cosets $r + I$ with well-defined addition and multiplication.

The even integers $2\mathbb{Z}$ form an ideal of $\mathbb{Z}$. The quotient $\mathbb{Z}/2\mathbb{Z}$ is the field with two elements.