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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.

## Examples and results

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