I am attempting to solve the following exercise:
Let $R$ be a ring, $I$ an ideal of $R$ and let $S$ be a semigroup. Show that the ring of semigroup $I[S]$ is an ideal of $R[S]$, and that $R[S]/I[S]$ is isomorphic to $(R/I)[S]$.
Well, first of all, I don't really understand what a semigroup ring is. My professor made the following definition:
Let $R$ be a ring and let $S$ be a multiplicative semigroup. The ring of semigroup $R[X]$ is the set of functions $S\rightarrow R$ of finite support for the following additive and multiplicative operations ($s\in S$): $$(f+g)(s)=f(s)+g(s)$$ $$(f*g)(s)=\sum_{s1s2=s}f(s1)g(s2)$$
I understand that these operatations were defined to add and multiply polynomials but I don't see why a semigroup ring is a set of finite support functions. Perhaps someone could provide me some intuitive pictures so that I may understand what I am dealing with.
Intuitively the functions are assigning coefficients to exponents. So if you have a function with infinite support then you have a "polynomial" with infinitely many terms. The semigroup ring is meant to have the universal property that, if you assign to X any value in an $R$-algebra $S$, then you get a map of $R$-algebras $R[X] \to S$ defined by plugging the value into each polynomial. But it's impossible in general to evaluate an infinite polynomial in an algebra because we only require them to have finite sums.
edit: actually it's a little more complicated than assigning values in an $R$-algebra since we are dealing with a semigroup instead of a group, but it's similar enough. The wikipedia article on this gives the precise definition.