## Ring Axioms

Definition. A ring is a set with the operations of addition and multiplication.

For any there is an element , so that R is closed under addition.

Addition is associative, i.e. for all There is an element of called the zero element and written 0,  which has the property that for all so that 0 is the identity element of the addition operation.

Every element has a negative, an element of written which satisfies so that the additive inverse of is These together imply that a ring is a group under addition with identity 0. for all This implies that a ring is an Abelian group under addition.

Multiplication

For any there is an element Multiplication is associative, i.e. for all Multiplication is distributive over addition, i.e. and for all A ring may or may not have a multiplicative identity. This means that the elements of the ring with the operation of multiplication may not necessarily be a group. Neither must the multiplication operation be commutative.

Examples: are all rings but the set of all ordered pairs with first component equal to 1 is not since and the first component is not 1 so the operation of addition is not closed. 