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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.)

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    $\begingroup$ If $r$ is invertible, this operation is just multiplication conjugated by the map $x\mapsto rx+1$. $\endgroup$ – Eric Wofsey Mar 10 '14 at 12:30
  • $\begingroup$ 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. $\endgroup$ – Jan Grabowski Mar 10 '14 at 12:42
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    $\begingroup$ Formal groups sprang to my mind. Maybe there are "formal monoids". $\endgroup$ – Torsten Schoeneberg Mar 10 '14 at 12:44
  • $\begingroup$ 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. $\endgroup$ – imakhlin Mar 10 '14 at 12:56
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The case $r=-1$ (or sometimes $r=1$) is the "circle operation" already used by Jacobson to define his radical in unitless rings.

A little google scholar finds

McConnell, N. R.; Stokes, T.: Generalising quasiregularity for rings. Austral. Math. Soc. Gaz. 25 (1998), 250-252

(and four papers citing it). The paper shows that among the "derived operations" (associative operations defined by noncommutative ring polynomial expressions) only seven kinds exists, the above one being the nontrivial one. It seeems of some use in the study of radical rings.

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