# Name and references for a “twisted” addition in a ring

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.)

• If $r$ is invertible, this operation is just multiplication conjugated by the map $x\mapsto rx+1$. – Eric Wofsey Mar 10 '14 at 12:30
• In the case of $\mathbb{Z}$ and $r=1$, one can also define a twisted addition by $a \oplus b=a+b+1$, making $\mathbb{Z}$ with $\oplus$ and your multiplication into a commutative ring. In fact, these operations are both just induced from the set bijection $a \mapsto a+1$ on the integers (so these two ring structures on $\mathbb{Z}$ are actually isomorphic) and @Eric Wofsey's comment shows how this generalises. – Jan Grabowski Mar 10 '14 at 12:42
• Formal groups sprang to my mind. Maybe there are "formal monoids". – Torsten Schoeneberg Mar 10 '14 at 12:44
• Yes, the case of an invertible $r$ is not too interesting. And the case of a general $r$ is probably best dealt with by localization. I'm really just looking for terminology and references. – imakhlin Mar 10 '14 at 12:56

The case $r=-1$ (or sometimes $r=1$) is the "circle operation" already used by Jacobson to define his radical in unitless rings.