Is there a meaningful bijection between tne set of all rings and the set of all groups? Thanks.

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Technically the classes of groups and rings are not sets, so there can be no bijection between them. But more to the point there is no natural *invertible* way to associate rings to groups. There are ways to associate a ring to each Abelian group (its ring of endomorphisms) and vice versa (its additive group), but each of these operations loses information, and also fails to produce every possible ring, respectively Abelian group (for the latter point see Does every Abelian group admit a ring structure?). So one cannot possibly hope to invert these operations.

In general it happens very rarely that two different kinds of structures turn out to be equivalent, and it is certainly not the case for groups and rings.

Marc van Leeuwen

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