Definition:Commutative and Unitary Ring

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Definition

A commutative and unitary ring $\struct {R, +, \circ}$ is a ring with unity which is also commutative.

That is, it is a ring such that the ring product $\struct {R, \circ}$ is commutative and has an identity element.

That is, such that the multiplicative semigroup $\struct {R, \circ}$ is a commutative monoid.

The identity element is usually denoted by $1_R$ or $1$ and called a unity.


Commutative and Unitary Ring Axioms

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.


Also known as

Other nomenclature includes:

  • Commutative and unital ring
  • Commutative ring with unity
  • Commutative ring with identity


Also see

  • Results about commutative and unitary rings can be found here.


Sources