Y by felixgotti
Let be a semiring, the Grothendieck group of is to be constructed by introducing inverse elements to all elements of . Elements of are of form with and in .
The multiplication of is extended to as follows:
which is well define since and are in .
As a hypothesis, is a semidomain; therefore, is a subsemiring of an integral domain . Then, the monoid is contained in the group . However, is the smallest group that contains , which means that is contained in .
According to this, is a subring of and therefore is an integral domain.
For the semiring , the multiplication of extends to turning into an integral domain. But is contained in , so is a subsemiring of . Thus, is a subsemiring of an integral domain. Consequently is a semidomain.