Leftins form a commutative ring

    Closure +*: a,bL_f , a+bL_f and abL_f .

a+b and ab are unique, thus they are binary operators on the leftins.

    Commutative +*: a,bL_f , a+b = b+a and ab = ba.

    Associative +*: a,bL_f , a+(b+c) = (a+b)+c and a(bc) = (ab)c.

    Distributive: a,b,cL_f , a(b+c) = ab + ac.

    Identity +: iL_f such that aL_f , a+i = a.
    Identity *: jL_f such that aL_f , aj = a.

Clearly ...(0).(0)...=0 serves as the additive identity while ...(0)1.(0)...=1 is the multiplicitive identity.

    Inverse +: aL_f , a'L_f such that a+a' = 0.

For example, if a = ...96318, a' = ...03682.

     L_f 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 +. L_f 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