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

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

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

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

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

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

Optional Properties

Optionally, a ring R may have additional properties:

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

Important Results

From the given axioms for a ring R , it can be deduced that:

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

Substructures, morphisms and quotients

Examples and results

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