← notes


ns-> Number Systems

Definition: A communative ring with 1 is a set $R$ having two (closed)[1] binary operations $+$ and $\times$ that satisfy eight axioms:

[A1] The operation $+$ is associative: $a+(b+c) = (a+b) + c$ for any elements $a,b,c \in R$

[A2] The operation $+$ is commutative: $a+b = b+a$ for any elements $a,b$ in $R$.

[A3] There is an additive identity $0$ satisfying $a+0 = a$ for all $a$ in R

[A4] Every element has an additive inverse $-a$ satisfying $a + (-a) = 0$

[A5] The operation $\times$ is associative

[1] The definition of a closed binary operation is the following: Let $S$ be a non-empty set. A binary operation $\times$ on $S$ is a closed binary operation if $a b \in S, \forall a,b \in S$.