# Definition:Commutative and Unitary Ring/Axioms

## Definition

A commutative and unitary ring is an algebraic structure $\left({R, *, \circ}\right)$, on which are defined two binary operations $\circ$ and $*$, which satisfy the following conditions:

 $(A0)$ $:$ Closure under addition $\displaystyle \forall a, b \in R:$ $\displaystyle a * b \in R$ $(A1)$ $:$ Associativity of addition $\displaystyle \forall a, b, c \in R:$ $\displaystyle \left({a * b}\right) * c = a * \left({b * c}\right)$ $(A2)$ $:$ Commutativity of addition $\displaystyle \forall a, b \in R:$ $\displaystyle a * b = b * a$ $(A3)$ $:$ Identity element for addition: the zero $\displaystyle \exists 0_R \in R: \forall a \in R:$ $\displaystyle a * 0_R = a = 0_R * a$ $(A4)$ $:$ Inverse elements for addition: negative elements $\displaystyle \forall a \in R: \exists a' \in R:$ $\displaystyle a * a' = 0_R = a' * a$ $(M0)$ $:$ Closure under product $\displaystyle \forall a, b \in R:$ $\displaystyle a \circ b \in R$ $(M1)$ $:$ Associativity of product $\displaystyle \forall a, b, c \in R:$ $\displaystyle \left({a \circ b}\right) \circ c = a \circ \left({b \circ c}\right)$ $(M2)$ $:$ Commutativity of product $\displaystyle \forall a, b \in R:$ $\displaystyle a \circ b = b \circ a$ $(M3)$ $:$ Identity element for product: the unity $\displaystyle \exists 1_R \in R: \forall a \in R:$ $\displaystyle a \circ 1_R = a = 1_R \circ a$ $(D)$ $:$ Product is distributive over addition $\displaystyle \forall a, b, c \in R:$ $\displaystyle a \circ \left({b * c}\right) = \left({a \circ b}\right) * \left({a \circ c}\right)$ $\displaystyle \left({a * b}\right) \circ c = \left({a \circ c}\right) * \left({b \circ c}\right)$

These criteria are called the commutative and unitary ring axioms.

These can be alternatively presented as:

 $(A)$ $:$ $\left({R, *}\right)$ is an abelian group $(M)$ $:$ $\left({R, \circ}\right)$ is a commutative monoid $(D)$ $:$ $\circ$ distributes over $*$