Now we come to a really nice example of a semigroup ring. Start
with the free commutative monoid on generators. This is just the
product of copies of the natural numbers: . Now let’s build the
semigroup ring on this monoid. First off, an element of the
monoid is an ordered -tuple of natural numbers . Let’s write it
in the following, more suggestive notation: . We multiply such
“monomials” just by adding up the corresponding exponents, as we
know from the composition rule for the monoid. Now we build the
semigroup ring by taking formal linear combinations of these
monomials. A generic element looks like where the are integers,
and all but finitely many of them are zero. Assuming everyone’s
taken high school algebra, we’ve seen these before. They’re just
polynomials in variables with integer coefficients! The addition
and multiplication rules are just what we know from high school
algebra. The only difference is here we specifically don’t think
of as a “placeholder” for a number, but as an actual element of
our