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 $B$?
Notation for the set of all isomorphisms between two modules.
2
$\begingroup$
abstract-algebra
notation
modules
-
0You can just identify them and write Aut. – 2012-05-03
-
3@Tobias: That requires you to commit yourself to a specific isomorphism identifying them. – 2012-05-03
-
4There should be no ambiguity with *isometries* so why not call it $\mathrm{Isom}_R( A,B)$? – 2012-05-03
-
0@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. – 2012-05-03
-
0@OlivierBégassat That was along the lines of what I was thinking of using; either $\operatorname{Iso}_{R}(A,B)$ or $\operatorname{Isom}_{R}(A,B)$. The latter in my opinion is preferable. – 2012-05-03
-
3@DavidWard: I prefer the former, because it sticks the the three-letter form of Hom, End and Aut. – 2012-05-03
-
0@TaraB From an aesthetic point of view, I agree with you! – 2012-05-03
-
0@DavidWard Please consider summarizing the comments into an answer, so that it gets removed from the [unanswered tab](http://meta.math.stackexchange.com/q/3138/15416). If you do so, it is helpful to post it to [this chat room](http://chat.stackexchange.com/rooms/9141/the-crusade-of-answers) to make people aware of it, so that it gets an upvote. For further reading upon the issue of too many unanswered questions, see [here](http://meta.stackexchange.com/q/143113/199957), [here](http://meta.math.stackexchange.com/q/1148/15416) or [here](http://meta.math.stackexchange.com/a/9868/15416). – 2013-06-08
1 Answers
3
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.