Irreducible elements and maximal ideals Let $A$ be some ring. It is well known that if $A$ is an integral domain, then an element $x$ is irreducible iff the principal ideal $xA$ is maximal in the set of principal ideals of $A$. Questions: is it true that 1) if $A$ is a factorial ring and $I$ an ideal of $A$, if it is maximal then it is generated by irreducible elements, 2) if $A$ is Noetherian (and an integral domain?), and $I$ an ideal of $A$, if it is maximal then it is generated by a finite number of irreducible elements.

More generally, if $A\,$ is a ring whose nonunits $\
eq 0\,$ factor into irreducibles then every prime ideal $\,P\
e 0\,$ in $A\,$ can be generated by irreducibles. If not, by Zorn the set of ideals $\,\subseteq P\,$ generable by irreducibles contains some $I$ maximal (in subset order). Since $\,I\
eq P\,$ there is some $\,a\in P\backslash I.\,$ By hypothesis $\, a = p_1\cdots p_n\,$ for irreducible $\,p_i,\,$ so $\,a\in P\,$ prime $\,\Rightarrow\,$ some $\,p_i\in P.\,$ Further $\,p_i\
ot\in I\,$ (else $\,a\in I),\,$ therefore $\,I\subsetneq I+(p_i)\subseteq P,\,$ contra maximality of $\,I.$