A ring $R$ is a triple $(R,+,\cdot)$ where $R$ is a nonempty set such that $(R,+)$ forms an abelian group, $(R,\cdot)$ forms a semigroup, and the two operations are related by the distributive laws: $a\cdot(b+c)=a\cdot b+a\cdot c$ and $(b+c)\cdot a=b\cdot a+c\cdot a$.
Important examples of rings include domains (such as the integers), fields (such as the real numbers), square matrix rings, polynomial rings, and rings of functions. Rings are studied in their own right in abstract algebra, but they are also prominently used in number theory, geometry, algebraic geometry, and logic.
The semigroup $(R,\cdot)$ is not always required to have an identity, but when it does, $R$ is called a ring with identity. For questions about rings which don't necessarily have a unit element, the tag rngs should be used.
The operation $\cdot$ does not have to be commutative, but when it is, $R$ is called a commutative ring.
There are numerous types of rings studied in different ways. An ideal in a ring is the ring-theortic analogue of a normal subgroup of a group. The study of ideals is an important component of ring theory.