# Notation for the set of all isomorphisms between two modules.

Let $R$ be an arbitrary ring. Given two $R$-modules $A$ and $B$, we may denote the set of all $R$-homomorphisms from $A$ to $B$ by $\operatorname{Hom}_{R}(A,B)$. If in addition we know that $A$ and $B$ are isomorphic $R$-modules, is there a specific way to denote the set of all $R$-isomorphisms from $A$ to $B$?

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You can just identify them and write Aut. –  Tobias Kildetoft May 3 '12 at 10:40
@Tobias: That requires you to commit yourself to a specific isomorphism identifying them. –  Chris Eagle May 3 '12 at 10:43
There should be no ambiguity with isometries so why not call it $\mathrm{Isom}_R( A,B)$? –  Olivier Bégassat May 3 '12 at 10:54
@Tobias Yes if you have a specific $R$-isomorphism, then we can form isomorphisms using the $R$-automorphisms of $A$ and $B$. However, in general this will not account for all such $R$-isomorphisms. –  David Ward May 3 '12 at 10:57
@DavidWard: I prefer the former, because it sticks the the three-letter form of Hom, End and Aut. –  Tara B May 3 '12 at 11:35

There doesn't seem to be a universally accepted notation, but the two that are clearly understood are $\operatorname{Iso}_{R}(A,B)$ and $\operatorname{Isom}_{R}(A,B)$. The former is probably preferable.

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