Question
I never understood why Sauron was not looking for the Ring actively in the Lord of the Rings films. The Nazgul know that Frodo has the Ring and they should be knowing where Frodo is heading.
Saruman has sent his orcs to find the Ring, but why was Sauron not that interested?
The Ring was not hidden. It was in the open, as Frodo had used the Ring multiple times. The fellowship was broken quickly and Frodo did not have any special power to hide the Ring.
Gandalf says that the ring wants to go to his master.
Is there any explanation of this in the books? It seems to me that the LotR movie series has omitted some important details about the relationship between the Ring and Sauron.
Answer
You'll find a lot of explanations on our sister site Science Fiction & Fantasy, e.g. Why didn't Sauron guard Mount Doom?. Those focus on the books, and most texts there didn't make the movies, but it can be explained from the movies alone, I think.
Basically, Sauron thinks everybody has the same intentions as he does, so the one wielding the Ring will use it to conquer Middle Earth. What feels more threatening, one or two little hobbits, or a combined army of Gondor and Rohan who just defeated Sauron's army at Minas Tirith? Plus, those hobbits could've joined the march to the Black Gate (they were last seen on the border between Gondor and Mordor, which is close enough to join). From Sauron's viewpoint that would make a lot of sense.
Plus, as @JoeL. puts it:
This was a key point of Gandalf and Aragorn's strategy, and the whole reason they led the army of Minas Tirith to the Black Gate of Mordor. The hope was to draw not only Sauron's attention, but his armies as well, leaving Mordor itself unguarded.
That's reflected in this scene after the battle for Minas Tirith:
Aragorn thinks Sauron will take the bait since he revealed himself as the heir to the throne of Gondor via the Palantir they snatched out of Orthanc.
Answered By - Glorfindel
