ring
A ring is a set where you can add, subtract, and multiply — but division is not promised. It abstracts the everyday arithmetic of the integers: you can add two whole numbers, take negatives, and multiply, yet 2 divided by 3 escapes the integers. A ring captures exactly “two interacting operations” without insisting that every nonzero element be invertible.
Formally, a ring is a set with two operations, addition and multiplication, where addition makes it an abelian group (with a zero element and negatives), multiplication is associative, and the two are tied together by the distributive law a · (b + c) = a · b + a · c (and on the other side too). Some authors also require a multiplicative identity 1; conventions vary, so check the source.
Multiplication in a ring need not be commutative — the ring of square matrices is a key non-commutative example. When multiplication does commute, the ring is called commutative, and that is the gateway to the special, well-behaved rings: an integral domain (no zero divisors) and, at the top, a field (every nonzero element invertible).
The integers ℤ form a commutative ring: addition gives an abelian group, multiplication is associative and commutative with identity 1, and distribution holds, e.g. 3 · (4 + 5) = 3·4 + 3·5 = 27. But ℤ is not a field, since 2 has no integer multiplicative inverse.
ℤ is a commutative ring but not a field.
Two facts that follow from the axioms alone: a · 0 = 0 for every element a, and (-a) · b = -(a · b). These hold even though “0” and negatives are defined purely through the additive structure — the distributive law forces multiplication to respect them.