No, algebraic integers are a different set than algebraic numbers. (A subset.)
Algebraic integers are much cooler, since there is a number theory on them: https://en.wikipedia.org/wiki/Algebraic_integer . (And also because the most basic facts about it, like that it forms a ring, are not trivial to prove, that's a good sign for a concept to be cool and useful.)
These two number sets are more or less in a relationship like regular integers (with a number theory), and rational numbers. In fact A = O/Z where A denotes the set of algebraic numbers, O denotes the set of algebraic integers, and Z denotes the set of integers.
Algebraic integers are much cooler, since there is a number theory on them: https://en.wikipedia.org/wiki/Algebraic_integer . (And also because the most basic facts about it, like that it forms a ring, are not trivial to prove, that's a good sign for a concept to be cool and useful.)
These two number sets are more or less in a relationship like regular integers (with a number theory), and rational numbers. In fact A = O/Z where A denotes the set of algebraic numbers, O denotes the set of algebraic integers, and Z denotes the set of integers.