### Leftins form a commutative ring

**Closure +*:**
a,b*L*_{f} ,
a+b*L*_{f} and
ab*L*_{f} .

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

**Commutative +*:**
a,b*L*_{f} ,
a+b = b+a and ab = ba.

**Associative +*:**
a,b*L*_{f} ,
a+(b+c) = (a+b)+c and a(bc) = (ab)c.

**Distributive:**
a,b,c*L*_{f} ,
a(b+c) = ab + ac.

**Identity +:**
i*L*_{f} such that
a*L*_{f} , a+i = a.

**Identity *:**
j*L*_{f} such that
a*L*_{f} , aj = a.

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

**Inverse +:**
a*L*_{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.

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© Scott Sherman 1999