In mathematics, a group is a set equipped with an associative binary operation (usually written as ·) which has an identity element (`e` such that `x·e = x = e·x`), in which every element `x` has an inverse `x^{-1}` such that `x · x^{-1} = e = x^{-1} x`.

In contrast, a ring is an abelian group (a group where `a + b = b + a`. In this case, the group operation is written as `+` instead of `·`, and `e` is called `0`) equipped with a second associative operation called multiplication. The symbol · usually denotes the ring multiplication operation, and + is used for the group operation.

Multiplication must have an identity element, usually called `1`, making a ring a monoid under multiplication.

Also, multiplication should distribute over group operations. `a · (b + c) = (a·b) + (b·c)` and `(b + c) · a = (b·a) + (c·a)`.

So, rings are groups with extra structure that allow them to support arithmetic.
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