Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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 $B$?

share|cite|improve this question
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
up vote 3 down vote accepted

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.