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Simple Problem concerning tensor products

Peter

Well-known member
MHB Site Helper
Jun 22, 2012
2,918
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

MHB Oldtimer
Staff member
Feb 7, 2012
2,725
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 $dx = (sm+tn)x = s(mx) + t(nx) = 0.$