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In a semmi-quasi local domain $D$ ( i.e. $D$ has finitely many maximal ideals),

an ideal $I$ is principal if and only if $ID_M$ is principal for all $M \in Max(D)$.

[See Comm. Rings by Kaplansky, (Theorem 60,61,62)

Is the result above could be true for a domain $D$ which is not a PID and having infinitely many maximal ideals?

In other words, Can we replace the semi-quasi-local condition with some other/weaker condition in above mentioned result?

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    $\begingroup$ I suggest you look up "factorial ring" / "unique factorization domain". $\endgroup$
    – Jason Starr
    Commented Aug 23, 2014 at 9:44

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A domain $D$ satisfies your condition if and only if it is an LPI domain with trivial Picard group (which can be realized as the group of invertible fractional ideals mod nonzero principal fractional ideals). This would include the case of UFDs. LPI domains are discussed, for example, in http://www.lohar.com/researchpdf/locallyprincipalisinvertible4.pdf and http://rms.unibuc.ro/bulletin/pdf/56-1/gabellis.pdf

Any Mori (hence any Noetherian) domain is an LPI domain, hence a Mori domain satisfies your condition if and only if it is has trivial Picard group. However, there exist Mori, and even Noetherian, domains with trivial class group that are not UFDs. See Seeking Noetherian normal domain with vanishing Picard group but not a UFD

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