# Standard method and name for extending a semiring to a ring

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?

• It depends on your semiring. In some cases, semirings do not obey the distributive law, and adding elements will not fix it, and often quotients also will not. Gerhard "Ask Me About System Design" Paseman, 2012.03.07 – Gerhard Paseman Mar 8 '12 at 1:23
• Thanks Gerhard, The version of semiring that I have been using is an additive commutative monoid and a multiplicative monoid related by the destributive law and 0 annihilates over multiplication. I would specify that n is a new element, commutative over R like a standard polynomial extension, though R not necessarily commutative. With distributivity specified, does the above extension and quotient of the semi-ring always work, or are there other problems? Also it seems like creating additive inverses should have a proper name ... – Richard Mar 8 '12 at 4:00
• Even if it is distributive, you are right that it can't always work. I can see how some semirings (like Max-plus) only can have inverses when quotiented with itself making a trivial ring. – Richard Mar 8 '12 at 13:38