Do I put a canonical tag on the page I am pointing to?
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Lets say B i a duplicate page of A (main page). I understand I have to put canonical tag under B to point to A. Do I also put canonical tag under the main page A? Is it necessary? I understand that A would then tell Google that it is preferred page of A? Is this a correct understanding?
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Best practice you should put the canonical tag onto the main page as well as the other pages.
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Yes, I'd also put the canonical tag on Page A referencing the Page A URL.
In general, self-referencing canonical tags are a good thing to implement site-wide. Then, when you do need to make a change and have one duplicate page's canonical tag point to another URL (in this case, Page B to Page A), the tag is already in place and you can make the change more easily.
But if self-referencing tags are in place from the beginning, they can also help squash lots of other duplicate content problems that end up occurring on more complex websites.
Hope that makes sense!