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My question is simple, though it proves to be much more difficult than it sounds. Suppose I want to find a binary operation to add extra structure to a multiplicative group (so it becomes a ring). This binary operation must be associative, and also distributive over multiplication by definition.

My question is whether or not there exists a binary operation that is both associative and distributive over multiplication. Keep in mind that I am just an amateur mathematician, not an advanced ring theorist, so it would be kind for you to keep things simple.

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A ring has a absorption element (zero) whereas a nontrivial group cannot have such an element. However sometimes it is possible to find an $R$ such that $R^\times=G$ for given groups $G$. But there will generally be more than one possible way to put a $+$ structure on $G\cup\{0\}$ (for example $\Bbb F_q^\times$ has many choices of generators), and I doubt there is a generic way to do it (I am not even sure if every group is a group of units). [It is true though that $G\subseteq R^\times$ is always solvable in $R$; just form a group ring.] –  anon Dec 30 '13 at 4:34
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Perhaps you are looking for en.wikipedia.org/wiki/Group_ring ? –  Prahlad Vaidyanathan Dec 30 '13 at 4:47
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You can only make it a ring if the multiplicative group is abelian. The distributivity of the binary operation somehow demands that. See math.stackexchange.com/q/609364/75923 and for a short answer to that question see math.stackexchange.com/a/609371/75923. –  drhab Dec 30 '13 at 10:11
    
@user118228 just to be absolutely clear, you're looking for an operation $\ast$ on a group $G$ such that $a\ast(bc)=(a\ast b)(a\ast c)$ and similarly for the right handed distributive law? I just have some lurking suspicions that you might not have expressed what you actually meant... –  rschwieb Dec 30 '13 at 14:32
    
@anon: math.stackexchange.com/questions/384422/… is a smaller version of your question –  Jack Schmidt Dec 30 '13 at 17:45
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