A ring is an algebraic structure comprised of a set paired with two operations on the set, which are designated as addition ${\displaystyle (+)}$ and multiplication ${\displaystyle (\cdot)}$ . As a group can be conceptualized as an ordered pair of a set and an operation, ${\displaystyle (G,\cdot)}$ , a ring can be conceptualized as an ordered triple ${\displaystyle (R, +, \cdot)}$ .

A set with addition and multiplication, ${\displaystyle (R, +, \cdot)}$ , is a field if and only if it satisfies the following properties:

1. Commutativity of addition — For all ${\displaystyle a,b\in R}$ , ${\displaystyle a + b = b + a}$
2. Associativity of both addition and multiplication — For all ${\displaystyle a,b,c\in R}$ , ${\displaystyle (a+b)+c=a+(b+c)}$ and ${\displaystyle (a\cdot b)\cdot c=a\cdot(b\cdot c)}$
3. Additive Identity — There exists a "zero" element, ${\displaystyle 0\in R}$ , called an additive identity, such that ${\displaystyle a+0=a=0+a}$ , for all ${\displaystyle a \in R}$
4. Additive Inverses — For each ${\displaystyle a \in R}$ , there exists a ${\displaystyle b\in R}$, called an additive inverse of ${\displaystyle a}$ , such that ${\displaystyle a+b=0}$
5. Distributive property — For all ${\displaystyle a,b,c\in R}$ , ${\displaystyle a\cdot(b+c)=a\cdot b+a\cdot c}$ and ${\displaystyle (b+c)\cdot a=b\cdot a+c\cdot a}$
6. Closure of addition and multiplication — For all ${\displaystyle a,b\in R}$ , ${\displaystyle a+b\in R}$ and ${\displaystyle a\cdot b\in R}$

A prototype of these kind of structures in the ring of the integers, ${\displaystyle (\Z,+,\cdot)}$ .

We will often abbreviate the multiplication of two elements, ${\displaystyle a\cdot b}$ , by juxtaposition of the elements, ${\displaystyle ab}$ . Also, when combining multiplication and addition in an expression, multiplication takes precedence over addition unless the addition is enclosed in parenthesis. That is, ${\displaystyle a+bc+d=a+(bc)+d}$ .

We can also denote ${\displaystyle -a}$ as the additive inverse of any ${\displaystyle a \in R}$ . Furthermore, we can define an operation, called subtraction by ${\displaystyle a-b=a+(-b)}$ .

Optional Properties

Optionally, a ring ${\displaystyle R}$ may have additional properties:

• We define ${\displaystyle R}$ to be a commutative ring if the multiplication is commutative: ${\displaystyle a\cdot b=b\cdot a}$ for all ${\displaystyle a,b\in R}$
• We define ${\displaystyle R}$ to be a ring with unity if there exists a multiplicative identity ${\displaystyle 1\in R}$ : ${\displaystyle 1\cdot a=a=a\cdot1}$ for all ${\displaystyle a \in R}$
  • Furthermore, a commutative ring with unity ${\displaystyle R}$ is a field if every element except 0 has a multiplicative inverse: For each non-zero ${\displaystyle a \in R}$ , there exists a ${\displaystyle b\in R}$ such that ${\displaystyle a\cdot b=b\cdot a=1}$
• We define ${\displaystyle R}$ to be an integral domain if it is commutative, has unity, and the zero product rule holds: ${\displaystyle a\cdot b=0}$ implies either ${\displaystyle a = 0}$ or ${\displaystyle b=0}$ for all ${\displaystyle a,b\in R}$ .

Important Results

From the given axioms for a ring ${\displaystyle R}$ , it can be deduced that:

• ${\displaystyle (R, +)}$ is an abelian group
• ${\displaystyle 0\cdot a=0}$ , for all ${\displaystyle a \in R}$
• ${\displaystyle a(-b)=(-a)b=-(ab)}$ , for all ${\displaystyle a,b\in R}$
• ${\displaystyle (-a)(-b)=ab}$ , for all ${\displaystyle a,b\in R}$
• If ${\displaystyle R}$ has unity ${\displaystyle 1}$
  • ${\displaystyle (-1)\cdot a=-a}$ , for all ${\displaystyle a \in R}$
  • ${\displaystyle (-1)\cdot(-1)=1}$
• Multiplication distributes over subtraction.

Substructures, morphisms and quotients

• subring
• ideal
• prime ideal
• maximal ideal
• field
• radical
• nilradical
• ring homomorphism
• quotient ring
• fundamental theorems

Examples and results

• integers
• chinese remainder theorem
• gaussian integers
• polynomial ring
• integral domain
• euclidean ring
• unique factorization domain
• non commutative ring