Let $R$ be a commutative and associative ring with unit. Can $R$ be embedded in a ring $\hat{R}$ wich is both non commutative and non associative ?
Thanks guys !
Let $R$ be a commutative and associative ring with unit. Can $R$ be embedded in a ring $\hat{R}$ wich is both non commutative and non associative ? Thanks guys ! 


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Yes. The "can" question is not so interesting: think of the image of the integers inside any nonassociative algebra. But specific cases, and counting embeddings, are interesting. See work of Gross and Gan, "Commutative Subrings of Certain Nonassociative Rings", Math. Ann. v.314 n.2, 1998. 

