example of tensor product of three finitely generated R-modules that equal 0

Question

Can you give an example of a commutative unitary ring $R$ and three finitely generated $R$-modules $L$, $M$ and $N$ such that $L\otimes M\ne 0$, $M\otimes N\ne 0$, and $L\otimes N\ne 0$, but $L\otimes N\otimes N = 0$?

Answer 1

We have $\mathbb{Z}/10\otimes_{\mathbb{Z}} \mathbb{Z}/15\neq 0$, $\mathbb{Z}/6\otimes_{\mathbb{Z}} \mathbb{Z}/10\neq 0$, $\mathbb{Z}/6\otimes_{\mathbb{Z}} \mathbb{Z}/15\neq 0$, but $\mathbb{Z}/6\otimes_{\mathbb{Z}} \mathbb{Z}/10 \otimes_{Z}\mathbb{Z}/15=\mathbb{Z}/2\otimes_{\mathbb{Z}} \mathbb{Z}/15=0$.

Answer 2

So you're asking for $L,M,N$ such that $$ L\otimes_R M \ne 0 \\ M\otimes_R N \ne 0 \\ N\otimes_R L \ne 0 $$ but $L\otimes_R M\otimes_R N=0$?

Mariano Suárez-Alvarez's hint is the way to go. Find integers $l,m,n$ such that $(l,m)$, $(m,n)$, and $(n,l)$ are larger than one but for which $(l,m,n) = 1$. The easiest way to do this is to choose distinct primes $p_i, 1\leqslant i\leqslant 3$ and put $$ l = p_2 p_3 \\ m = p_1 p_3 \\ n = p_1 p_2 . $$ Then $L=\mathbf Z/l$, $M=\mathbf Z/m$, and $N=\mathbf Z/n$ have the desired properties.