Ring

The term ring is used in four different senses:


 * Ring which may not be commutative and may not have a multiplicative identity
 * Unital ring which is a ring with multiplicative identity
 * Commutative ring which is a ring where the multiplication operation is commutative
 * Commutative unital ring which is both a commutative ring and a unital ring

This article gives the first definition

Symbol-free definition
A ring is a set with two structures, addition and multiplication such that it forms an Abelian group under addition and a semigroup under multiplication, and such that multiplication satisfies both left distributivity and right distributivity over addition.

Definition with symbols
A ring is a set $$R$$ endowed with a constant $$0$$, a unary operation $$-$$ and binary operations $$+$$ and $$*$$ such that:


 * $$a + (b + c) = (a + b) + c$$ for all $$a, b, c$$ in $$F$$ (associativity of addition)
 * $$a + 0 = a$$ for all $$a$$ in $$F$$ (additive neutral element)
 * $$a + b = b + a$$ for all $$a, b$$ in $$F$$ (commutativity of addition)
 * $$a + (-a) = 0$$ for all $$a$$ in $$F$$ (inverse for addition)
 * $$a * (b * c) = (a * b) * c$$ for all $$a, b, c$$ in $$F$$ (associativity of multiplication)
 * $$a * (b + c) = (a * b) + (a * c)$$ for all $$a, b, c$$ in $$F$$
 * $$(a + b) * c = (a * c) + (b * c)$$ for all $$a, b, c$$ in $$F$$

Homomorphism of rings
A ring homomorphism is a function from one ring to another that maps 0 to 0, and also preserves the unary $$-$$ operation and the binary operations $$+$$ and $$*$$.

Subring
A subring is a subset of a ring that is a closed under the ring operations and hence forms a ring by restricting these operations to it.