I'm doing some algebra and I don't know if a construction I'm doing already has a standard name. It has to do with a generalization of power series with an arbitrary number of variables. The construction is as follows.
Let $R$ be a commutative ring with $1$ and let $G$ be a monoid with the following property: for each $u\in G$, the set of pairs $(v,w)\in G^2$ such that $v+w=u$ is finite. We then can form the following rings
$R[[G]]$ is the set of all functions $f:G\to R$. Addition and multiplication in $R[[G]]$ is defined as follows
\begin{align*} (f+g)(u)&=f(u)+g(u) \\ (fg)(u)&=\sum_{v+w=u}f(v)g(w) \end{align*}
Then $R[[G]]$ becomes a commutative ring with $1$ (but I don't know if this construction has a name).
$R[G]$ is the subring of $R[[G]]$ of functions $f:G\to R$ such that $f(u)=0$ for all but a finite number of $u$'s. $R[G]$ is called the monoid ring of $G$ over $R$.
When $G=\mathbb{N}^{\oplus I}$, $R[[G]]$ is exactly the ring of power series with $I$ variables $X_i$ and $R[G]$ is exactly the ring of polynomials with $I$ variables.
I'm working with $R[[G]]$ but I don't know if it already has a standard name. I've already looked at several books and they only define $R[G]$. For example, Serge Lang's algebra book does these constructions (though he is not as explicit as I am) in page 106.