It is decades since I've done math, so please forgive the lack of correct terminology and lack of latex etc. I'm endeavoring to write a simple CAS calculator that can handle structures that undergraduates could run into.


Thanks to wikipedia, I've found that the way of adding a multiplicative inverse to a ring to create a field is called 'localization' - 
where R is the ring, S = R - zero divisors of R, 


the new field is basically an ordered pair (S, R), with the meaning S^-1 \* R
I guess there is an implied cancellation law added so that (x,x) => (1,1), along with standard rules for rational addition, multiplication, and inverse.


Have I got this right so far?


Anyway, I figure that creating an additive inverse for a semiring is
R[n], where n^2 = 1, and x+xn=0, n is the symbol for negative.
Is this the quotient group R[n]/(n^2 - 1, 1 + n)?


Is there any proper name for this process, and do I have it correct?



Standard method and name for extending a semiring to a ring
I think it is often called forming the Grothendieck ring. The construction is really defined for commutative monoids, where one calls it the Grothendieck group and then you carry the multiplication along for the ride in the semiring case. Peter May uses this terminology on page 199 of his concise course on algebraic topology. 


