Example of $\sum_i a_i\otimes b_i\in M\otimes_AN$ which cannot be written as $a\otimes b$

Question

In the appendix of my commutative algebra text:

Note that in general the element of $M\otimes_AN$ is a sum of the form $\sum_i a_i\otimes b_i$ and cannot be necessarily written as $a\otimes b$.

Can anyone provide simple counterexample to support the statement?

Answer 1

Take $M=N=\mathbb Z[X]$, which is a $\mathbb Z$-module. Then you cannot write $1\otimes_\mathbb Z X+X\otimes_\mathbb Z 1$ as $a(X)\otimes_\mathbb Z b(X)$. Henceforth take $\otimes=\otimes_\mathbb Z$.

If we try, we see that $a$ and $b$ cannot have degree more than $1$, else we introduce an $X^2$ which we cannot get rid of. Hence we rewrite our situation $aX+b\otimes cX+d$. We can expand this to obtain $$ aX\otimes cX+aX\otimes d+b\otimes cX+b\otimes d. $$ The first term doesn't work, since it has an $X$ on both sides, so $a=0$ or $c=0$. The last term doesn't work since it doesn't have an $X$, so $b=0$ or $d=0$. In any of these cases, you end up with one term which is not equal to the term we started with.