Closure +*: a,bLf , a+bLf and abLf .
a+b and ab are unique, thus they are binary operators on the leftins.
Commutative +*: a,bLf , a+b = b+a and ab = ba.
Associative +*: a,bLf , a+(b+c) = (a+b)+c and a(bc) = (ab)c.
Distributive: a,b,cLf , a(b+c) = ab + ac.
iLf such that
aLf , a+i = a.
Identity *: jLf such that aLf , aj = a.
Clearly ...(0).(0)...=0 serves as the additive identity while ...(0)1.(0)...=1 is the multiplicitive identity.
Inverse +: aLf , a'Lf such that a+a' = 0.
For example, if a = ...96318, a' = ...03682.
Lf is a commutative ring over the binary operators + and *.
A commutative ring requires two binary operators that are commutative, associative, distributive, and there is an identity and inverse for +. Lf satisfies these properties.
The questions of whether a multiplicitive identity exists or divisors of 0 exist will be addressed in the section fields and integral domains.
© Scott Sherman 1999