Are people born in Puerto Rico U.S Citizens?
Yep, people born in Puerto Rico are US citizens. So are people born in Guam or the US Virgin Islands or any of the other territories of the US. Regardless of that, since the mother was born in the US, she’s an American citizen and would pass her citizenship down to her children no matter where they were born.
Puerto Rico is part of the USA, therefore all residents born in PR are US citizens and have the right to live in the US mainland without any immigration process or restriction and hold a US passport. They have all of the rights as US citizens, except that a residency in PR will not grant you the right to vote in national elections because PR is not a state. It doesn’t matter if you were born on the mainland or not because even those born on the mainland who moved to PR will lose their right to vote in national elections until they ever move back to a state.