Commutative rings and algebras are sets like the set of integers, allowing addition and (commutative) multiplication. Of particular interest are several classes of rings of interest in number theory, field theory, algebraic geometry, and related areas; however, other classes of rings arise, and a rich structure theory arises to analyze commutative rings in general, using the concepts of ideals, localizations, and homological algebra.

A commutative ring is a set endowed with two binary operations "+" and "*" subject to familiar associative, commutative, and distributive laws. (It is usually but not universally assumed that the rings contain an identity element "1" for multiplication.) Examples include the rings of integers in algebraic number fields; here, the interest is number-theoretic: common questions concern factorization and the class group, the action of the Galois group, and the structure of the group of units. A commutative algebra is a commutative ring which contains a field (usually as a subring over which the entire ring is finitely-generated). Examples include coordinate rings of algebraic varieties, that is, quotients of polynomial rings over a field; here, the interest is geometric: how are the local rings different at singular points, and how do subvarieties intersect?