  Given a ring homomorphism $B \to A$, consider its centralizer $R = A^B$,
bimodule endomorphism ring $S = \End {}_BA_B$ and sub-tensor-square ring $T =
(A \o_B A)^B$. Nonassociative tensoring by the cyclic modules $R_T$ or ${}_SR$
leads to an equivalence of categories inverse to the functors of induction of
restricted $A$-modules or restricted coinduction of $B$-modules in case $A \|
B$ is separable, H-separable, split or left depth two (D2). If $R_T$ or ${}_SR$
are projective, this property characterizes separability or splitness for a
ring extension. Only in the case of H-separability is $R_T$ a progenerator,
which replaces the key module $A_{A^e}$ for an Azumaya algebra $A$. After
establishing these characterizations, we characterize left D2 extensions in
terms of the module $T_R$, and ask whether a weak generator condition on $R_T$
might characterize left D2 extensions as well, possibly a problem in
$\sigma(M)$-categories or its generalizations.
  We also show that the centralizer of a depth two extension is a normal
subring in the sense of Rieffel as well as pre-braided commutative. For
example, its normality yields a Hopf subalgebra analogue of a factoid for
subgroups and their centralizers, and a special case of a conjecture that D2
Hopf subalgebras are normal.
