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

A set with addition and multiplication, , is a field if and only if it satisfies the following properties:

  1. Commutativity of addition — For all ,
  2. Associativity of both addition and multiplication — For all , and
  3. Additive IdentityThere exists a "zero" element, , called an additive identity, such that , for all
  4. Additive Inverses — For each , there exists a , called an additive inverse of , such that
  5. Distributive property — For all , and
  6. Closure of addition and multiplication — For all , and

A prototype of these kind of structures in the ring of the integers, .

We will often abbreviate the multiplication of two elements, , by juxtaposition of the elements, . Also, when combining multiplication and addition in an expression, multiplication takes precedence over addition unless the addition is enclosed in parenthesis. That is, .

We can also denote as the additive inverse of any . Furthermore, we can define an operation, called subtraction by .

Optional Properties

Optionally, a ring may have additional properties:

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

Important Results

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

  • is an abelian group
  • , for all
  • , for all
  • , for all
  • If has unity
    • , for all
  • Multiplication distributes over subtraction.

Substructures, morphisms and quotients

Examples and results

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