What does the $(m,n)$ mean in context of $\mathbb{Z}_{(m,n)}$

Question

On the section about the Hom sets of modules, Hungerford has an exercise that asks to show that $$\operatorname{Hom}(\mathbb{Z}_m, \mathbb{Z}_n) \cong \mathbb{Z}_{(m,n)}$$ and then in the next exercise he has

If $A,B$ are abelian groups and $m,n$ integers such that $mA = 0 = nB$, then every element of $\operatorname{Hom}(A,B)$ has order dividing $(m,n)$.

What does $(m,n)$ mean in these contexts?

In number theory $,(a,b),$ is commonly used to denote $,\gcd(a,b),$ (same as the ideal $,(a,b) = (a) + (b) $ in PIDs, so one can give unified proofs for ideals and gcds using this notation) Also $,[m,n] = (m)\cap (n), $ is used for lcm, though that is less common — Bill Dubuque, Jul 27 '16 at 18:15

score 1 · Accepted Answer · answered Jul 27 '16 at 18:08

1

It means the greatest common divisor of $m$ and $n$.

answered Jul 27 '16 at 18:08

Michael Albanese

99,526

What does the $(m,n)$ mean in context of $\mathbb{Z}_{(m,n)}$

1 Answers1