Definition: a ring $R$ satisfies the ascending chain condition (ACC) on principal ideals if given a sequence $\langle a_1 \rangle \subset \langle a_2 \rangle \subset...\subset \langle a_n \rangle \subset ......$ $\exists N$ such that $\langle a_n \rangle = \langle a_{n+1} \rangle$ $\forall n\ge N$.
Proposition: For an integral domain R, the following are equivalent: (i) R satisfies the ascending chain condition on principal ideals. (ii) Every non-empty collection of principal ideals has a maximal element.
I think that we need to update the proposition to say: Proposition: For an integral domain R, the following are equivalent: (i) R satisfies the ascending chain condition on principal ideals. (ii) Every non-empty collection of proper principal ideals has a maximal element.
Is this correct? We could have a non empty set $X$ of units, and then for all $x \in X$ we have $xR=R$ and so there is no maximal element.
An element $m$ of a partially ordered set $(X,\leq)$ is called maximal if there are no elements greater than $m$, that is, if $m\leq a$, then also $a\leq m$. In other words, if an element is greater than or or of equal size as $m$, then it is of equal size, not greater than.
An ideal $\mathfrak m\subsetneq R$ of a ring $R$ is called maximal if there is no proper ideal of $R$ containing $\mathfrak m$, that is, for any ideal $\mathfrak a\subseteq R$ it holds that $\mathfrak m\subseteq\mathfrak a~\Rightarrow~\mathfrak a=\mathfrak m$ or $\mathfrak a=R$.
While these notions are related (maximal ideals are the maximal elements of the set of proper ideals partially ordered by the subset relation), they are not synonymous. There are partially ordered sets of ideals which have a maximal element, but contain no maximal ideals. Any partially ordered set containing a single ideal, for instance, has a maximal element. This is because in any partially ordered set containing a single element, that element is a maximal element. It's the only element, so there obviously can't be a larger one. And the other way around is also possible: A set of ideals partially ordered by the subset relation may contain maximal ideals without those being maximal elements of the partially ordered set. This is possible if the whole ring is included in the set.