An algebra extension A| B is right depth two in this paper if its tensor-square is A-B-isomorphic to a direct summand of any ( not necessarily finite) direct sum of A with itself. For example, normal subgroups of infinite groups, infinitely generated Hopf-Galois extensions and infinite-dimensional algebras are depth two in this extended sense. The added generality loses some duality results obtained in the finite theory ( Kadison and Szlachanyi, 2003) but extends the main theorem of depth two theory, as for example in ( Kadison and Nikshych, 2001). That is, a right depth two extension has right bialgebroid T = ( A circle times(B) A)(B) over its centralizer R = C-A(B). The main theorem: An extension A| B is right depth two and right balanced if and only if A| B is T-Galois with respect to left projective, right R-bialgebroid T.
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