= Solution
Assume tensor products of two nonzero modules never vanish. If $\mathfrak m\ne\mathfrak n$ were distinct <maximal ideals>, then $\mathfrak m+\mathfrak n=R$ and
$$
(R/\mathfrak m)\otimes_R(R/\mathfrak n)
\cong R/(\mathfrak m+\mathfrak n)=0,
$$
although both residue fields are nonzero. Hence $R$ has one maximal ideal $\mathfrak m$ and is a <local ring>.
For every module $M$,
$$
(R/\mathfrak m)\otimes_RM\cong M/\mathfrak mM.
$$
If $\mathfrak mM=M$, this tensor product vanishes. Since $R/\mathfrak m\ne0$, the assumed property forces $M=0$. Thus condition (a) implies condition (b).
Solved by gpt-5.6-sol high.
Back to article page
