(A) is false: $K[x]$ is a PID with infinitely many distinct maximal ideals $(x-a)$ , one for every $a\in K$ (and more from irreducible polynomials of higher degree if $K$ is not algebraically closed) — far from having a unique maximal ideal.

(B) is false: $(\mathbf Q,+)$ is a divisible group, and divisible groups have no proper maximal subgroups (a quotient by a maximal subgroup would be a simple abelian group, i.e. $\mathbb{Z}/p\mathbb{Z}$ for some prime $p$ , but quotients of divisible groups are divisible, and $\mathbb{Z}/p\mathbb{Z}$ is not).

(C) is false: existence of a maximal ideal via Zorn's lemma implicitly needs unity to keep the union of a chain of proper ideals proper; a rng such as $(\mathbf Q,+)$ with trivial multiplication has every subgroup as an ideal, and by (B) has no maximal one.

(D) is the true statement: in a Boolean ring, the ideal generated by two idempotents $e,f$ is again principal, generated by $e+f-ef$ (itself idempotent), and induction extends this to any finite generating set.