Simple Problem concerning tensor products

[Math Amateur]

Actually this problem really only concerns greatest common denominators.

In Section 10.4, Example 3 (see attachment) , Dummit and Foote where we are dealing with the tensor product \(\displaystyle \mathbb{Z} / m \mathbb{Z} \otimes \mathbb{Z} / n \mathbb{Z}\) we find the following statement: (NOTE: d is the gcd of integers m and n)

"Since \(\displaystyle m(1 \otimes 1) = m \otimes 1 = 0 \otimes 1 = 0 \)

and similarly

\(\displaystyle n(1 \otimes 1) = 1 \otimes n = 1 \otimes 0 = 0 \)

we have

\(\displaystyle d(1 \otimes 1) = 0 \) ... ... "

Basically we have

mx = 0 and nx = 0 and have to show dx = 0

It must be simple but I cannot see it!

Can someone please help?

Peter

 

[Opalg]

This comes from the fact that the gcd of two numbers can always be written as a linear combination of them. If $d = \text{gcd}(m,n)$ then there are integers $s,t$ such that $d=sm+tn.$ Then $dx = (sm+tn)x = s(mx) + t(nx) = 0.$