An algebra extension A | B is right depth two if its tensor-square A ⊗ B A is in the Dress category Add A A B . We consider necessary conditions for right, similarly left, D2 extensions in terms of partial A-invariance of two-sided ideals in A contracted to the centralizer. Finite dimensional algebras extending central simple algebras are shown to be depth two. Following P. Xu, left and right bialgebroids over a base algebra R may be defined in terms of anchor maps, or representations on R. The anchor maps for the bialgebroids S = End B A B and T = End A A ⊗ B A A over the centralizer R = C A (B) are the modules S R and R T studied in Kadison (J. Alg. & Appl., 2005, preprint), Kadison (Contemp. Math., 391: 149-156, 2005), and Kadison and Külshammer (Commun. Algebra, 34: 3103-3122, 2006), which provide information about the bialgebroids and the extension (Kadison, Bull. Belg. Math. Soc. Simon Stevin, 12: 275-293, 2005). The anchor maps for the Hopf algebroids in Khalkhali and Rangipour (Lett. Math. Phys., 70: 259-272, 2004) and Kadison (2005, preprint) reverse the order of right multiplication and action by a Hopf algebra element, and lift to the isomorphism in Van Oystaeyen and Panaite (Appl. Categ. Struct., 2006, in press). We sketch a theory of stable A-modules and their endomorphism rings and generalize the smash product decomposition in Kadison (Proc. Am. Math. Soc., 131: 2993-3002, 2003 Prop. 1.1) to any A-module. We observe that Schneider's coGalois theory in Schneider (Isr. J. Math., 72: 167-195, 1990) provides examples of codepth two, such as the quotient epimorphism of a finite dimensional normal Hopf subalgebra. A homomorphism of finite dimensional coalgebras is codepth two if and only if its dual homomorphism of algebras is depth two.