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 $F$

Important notions

Homomorphism of rings

Further information: Ring homomorphism

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 $*$.

Ideal

Further information: Ideal

Subring

Further information: 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.