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 $\struct {R, *, \circ}$, on which are defined two binary operations $\circ$ and $*$, which satisfy the following conditions:

\((\text A 0)\)   $:$   Closure under addition      \(\displaystyle \forall a, b \in R:\) \(\displaystyle a * b \in R \)             
\((\text A 1)\)   $:$   Associativity of addition      \(\displaystyle \forall a, b, c \in R:\) \(\displaystyle \paren {a * b} * c = a * \paren {b * c} \)             
\((\text A 2)\)   $:$   Commutativity of addition      \(\displaystyle \forall a, b \in R:\) \(\displaystyle a * b = b * a \)             
\((\text A 3)\)   $:$   Identity element for addition: the zero      \(\displaystyle \exists 0_R \in R: \forall a \in R:\) \(\displaystyle a * 0_R = a = 0_R * a \)             
\((\text A 4)\)   $:$   Inverse elements for addition: negative elements      \(\displaystyle \forall a \in R: \exists a' \in R:\) \(\displaystyle a * a' = 0_R = a' * a \)             
\((\text M 0)\)   $:$   Closure under product      \(\displaystyle \forall a, b \in R:\) \(\displaystyle a \circ b \in R \)             
\((\text M 1)\)   $:$   Associativity of product      \(\displaystyle \forall a, b, c \in R:\) \(\displaystyle \paren {a \circ b} \circ c = a \circ \paren {b \circ c} \)             
\((\text M 2)\)   $:$   Commutativity of product      \(\displaystyle \forall a, b \in R:\) \(\displaystyle a \circ b = b \circ a \)             
\((\text M 3)\)   $:$   Identity element for product: the unity      \(\displaystyle \exists 1_R \in R: \forall a \in R:\) \(\displaystyle a \circ 1_R = a = 1_R \circ a \)             
\((\text D)\)   $:$   Product is distributive over addition      \(\displaystyle \forall a, b, c \in R:\) \(\displaystyle a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c} \)             
\(\displaystyle \paren {a * b} \circ c = \paren {a \circ c} * \paren {b \circ c} \)             


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