Infinite index subalgebras of depth two

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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 math.RA/0108067 but extends the main theorem of depth two theory, as for example in math.RA/0107064. That is, a right depth two extension has right bialgebroid T = (A ⊗B A)ᵇ over its centralizer R = Cₐ(B). The main theorem: an extension A | B is right depth two and right balanced if and only if A | B is T-Galois wrt. left projective, right R-bialgebroid T.