Given an infinite distributive lattice $L$, does $L$ contain a prime ideal $I$ that is non-principal? ($I$ is said to be principal if there is $x\in L$ such that $I=\{y\in L: y\leq x\}$.)

If the answer is "yes" to the question above, what if we replace "prime" by "maximal"?

Given an infinite distributive lattice $L$, does $L$ contain a non-principal prime ideal $I$, or a non-principal prime filter $F$? ($I$ is said to be principal if there is $x\in L$ such that $I=\{y\in L: y\leq x\}$. Dual definition for filters.)

If the answer is "yes" to the question above, what if we replace "prime" by "maximal"?