SCHWEITZER 1978 Problem 2

For a distributive lattice $L$ , consider the following two statements:

(A) Every ideal of $L$ is the kernel of at least two different homomorphisms.

(B) $L$ contains no maximal ideal.

Which one of these statements implies the other?

(Every homomorphism $\varphi$ of $L$ induces an equivalence relation on $L$ : $a \sim b$ if and only if $a \varphi=b \varphi$ . We do not consider two homomorphisms different if they imply the same equivalence relation.)

J. Varlet, E. Fried