Applications of commutative semigroup rings

Let $R$ be a commutative ring. Then every commutative semigroup $S$ is naturally associated to a commutative ring extension of $R$, the commutative semigroup ring of $R$ by $S$. It is not hard to see that this construction is full of applications in commutative algebra and algebraic geometry. We will utilize commutative semigroup rings as an organizing principle in the theory of polynomial rings, which is an important part of algebraic geometry.

Polynomial rings:

The free $\mathbb{N}$-semimodule $F(X)$ is a very familiar object of commutative semigroup theory. It is not hard to see that the polynomial ring $R[x_1,x_2,...]$ is merely the commutative semigroup ring of $R$ by $F(x_1,x_2,...)$ : $RF(x_1,x_2,..)$. As a consequence, the polynomial rings that are so fundamental in algebraic geometry, can be considered to be a special case of a commutative semigroup ring.

Subalgebras of polynomial rings

Let $S$ be a finitely generated torsion-free cancellative J-trivial commutative semigroup. Then $S$ embeds into the free commutative semigroup $F(X)$ on a finite set of generators $X$. As a consequence, we can embed the commutative semigroup ring $RS$ into the polynomial ring $R[x_1,x_2,...]$.

As an example, any numerical semigroup can be embedded in the polynomial ring on a single generator. The polynomial subring $R[x^2,x^3]$ for example is merely the commutative semigroup ring of the numerical semigroup $\{2,3\}$. If we had $R[x^2y,yz^3]$ for example it would be generated by the commutative semigroup $(x^2y,yz^3) \in F(x,y)$, and so on.

Extensions of polynomial rings:

It is a basic fact of commutative algebra that $F(X)$ is a cancellative semigroup. Therefore, the free $\mathbb{N}$ semimodule $F(X)$ can be embedded in the free $\mathbb{Z}$-module $F^{\circ}(X)$. As a consequence, the commutative semigroup ring of multivariable polynomials $RF(X)$ can be embedded in the ring of multivariable Laurent polynomials $RF^{\circ}(X)$.

This can be further extended by considering rings of Puiseux polynomials $R\mathbb{Q}^n$ consisting of polynomials that have rational exponents, or this could even be embedded in $R\mathbb{R}^n$ to have arbitrary real exponents, so that we can have a complete extension of the ordinary polynomial ring $R[x_1,x_2,...]$.

Coordinate rings of varieties

Let $Y$ be an algebraic variety defined by a system of polynomial equations in $R[x_1,x_2,...]$. Then by now means is it the case that the coordinate ring $A(Y)$ can always be defined by a commutative semigroup ring. However, there is an important case in which they can be: algebraic varieties defined by differences of monomials. These correspond to relations in the presentation of a commutative semigroup.

Therefore, we can use commutative semigroup rings in algebraic geometry in order to deal with the important special case of varieties determined by differences of monomials. For example, consider the hyperbola $\frac{R[x,y]}{xy=1}$. Then this clearly produces a presentation of the commutative group $\mathbb{Z}$ so this is a ring of Laurent polynomials. As you can see, this is a very useful concept of commutative algebra.

References:
[1] Commutative semigroup rings by Gilmer