A commutative ring $R$ has finite character if every nonzero element of $R$ is contained in at most finitely many maximal ideals. I am interested in the particular case in which $R$ is a Noetherian domain. I don't know a lot about this case other than that Dedekind domains have finite character, as well as $1$-dimensional Noetherian domains. What I am hoping is that this class of rings (Noetherian domains of finite character) is fairly robust. If someone could supply an example or two to show what scope this class has, I would be grateful.
EDIT: This question arises from an expository paper I have been preparing. I have generalized arguments of Rotman and Enochs to construct co-Hopfian modules (over Noetherian rings of finite character) of uncountable cardinality that arise as direct products. Following an argument of user "moonlight" on this site, I have shown that the injective envelope of these modules is not co-Hopfian if the ring is also an integral domain (with certain cardinality requirements on its maximal spectrum) . What I am hoping, and have been unable to determine or find in the literature, is that my result is valid for more than $1$-dimensional Noetherian domains, hence my request for assistance.
EDIT 2: After poring over my copy of Kaplansky's Commutative Rings, I was wondering if there may be a connection to Krull domains.