[−][src]Trait mathru::algebra::abstr::Ring
Ring
Definition
- $
\mathbb{R}$ is an abelian group under addition, meaning that:
- $
(a + b) + c = a + (b + c), \forall a, b, c \in \mathbb{R}$ (that is,- is associative)
- $
a + b = b + a , \forall a, b \in \mathbb{R}$ (that is, + is commutative). - There is an element 0 in $
\mathbb{R}$ such that $a + 0 = a, \forall a \in \mathbb{R}$ (that is, 0 is the additive identity) - For each a in $
\mathbb{R}$ there exists −a in $\mathbb{R}$ such that $a + (−a) = 0$ (that is, −a is the additive inverse of a)
- $
\mathbb{R}$ is a monoid under multiplication, meaning that:
- $
(a * b) * c = a * (b * c), \forall a, b, c \in \mathbb{R}$ (that is, * is associative) - There is an element 1 in R such that $
a · 1 = a \wedge 1 · a = a, \forall a \in \mathbb{R}$ (that is, 1 is the multiplicative identity)
- Multiplication is distributive with respect to addition, meaning that:
- $
a * (b + c) = (a * b) + (a * c), \forall a, b, c \in \mathbb{R}$ (left distributivity) - $
(b + c) * a = (b * a) + (c * a), \forall a, b, c \in \mathbb{R}$ (right distributivity)