This question may seem a little unmotivated, but it actually isn't something that just occurred to me out of nowhere. It stems from a discussion I had the other day with a friend and is directly related to her research.
Consider a commutative ring $R$ and $r\in R$. Let us define $$a\#b=a+b+abr.$$ Shockingly, $\#$ is associative and commutative, i.e. defines a monoid structure on $R$. One may then proceed to study the group of elements invertible under $\#$ and what not.
Questions. Is there a proper name for such an operation (or some generalization thereof)? Where can one find a discussion of its and the above-mentioned group's properties? (In general or for specific rings.)