Axiom:Commutative and Unitary Ring/Axioms
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Definition
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 | \(\ds \forall a, b \in R:\) | \(\ds a * b \in R \) | |||||
\((\text A 1)\) | $:$ | Associativity of addition | \(\ds \forall a, b, c \in R:\) | \(\ds \paren {a * b} * c = a * \paren {b * c} \) | |||||
\((\text A 2)\) | $:$ | Commutativity of addition | \(\ds \forall a, b \in R:\) | \(\ds a * b = b * a \) | |||||
\((\text A 3)\) | $:$ | Identity element for addition: the zero | \(\ds \exists 0_R \in R: \forall a \in R:\) | \(\ds a * 0_R = a = 0_R * a \) | |||||
\((\text A 4)\) | $:$ | Inverse elements for addition: negative elements | \(\ds \forall a \in R: \exists a' \in R:\) | \(\ds a * a' = 0_R = a' * a \) | |||||
\((\text M 0)\) | $:$ | Closure under product | \(\ds \forall a, b \in R:\) | \(\ds a \circ b \in R \) | |||||
\((\text M 1)\) | $:$ | Associativity of product | \(\ds \forall a, b, c \in R:\) | \(\ds \paren {a \circ b} \circ c = a \circ \paren {b \circ c} \) | |||||
\((\text M 2)\) | $:$ | Commutativity of product | \(\ds \forall a, b \in R:\) | \(\ds a \circ b = b \circ a \) | |||||
\((\text M 3)\) | $:$ | Identity element for product: the unity | \(\ds \exists 1_R \in R: \forall a \in R:\) | \(\ds a \circ 1_R = a = 1_R \circ a \) | |||||
\((\text D)\) | $:$ | Product is distributive over addition | \(\ds \forall a, b, c \in R:\) | \(\ds a \circ \paren {b * c} = \paren {a \circ b} * \paren {a \circ c} \) | |||||
\(\ds \paren {a * b} \circ c = \paren {a \circ c} * \paren {b \circ c} \) |
These criteria are called the commutative and unitary ring axioms.
These can be alternatively presented as:
\((\text A)\) | $:$ | $\struct {R, *}$ is an abelian group | |||||||
\((\text M)\) | $:$ | $\struct {R, \circ}$ is a commutative monoid | |||||||
\((\text D)\) | $:$ | $\circ$ distributes over $*$ |