  We consider several natural ways of expressing the idea that a one-sided
ideal in a C*-algebra (or a submodule in a Hilbert C*-module) is large, and
show that they differ, unlike the case of two-sided ideals in C*-algebras. We
then show how these different notions, for ideals and for submodules, are
related. We also study some permanence properties for these notions. Finally,
we use essential right ideals to extend the inner product on a Hilbert
C*-module to a part of the dual module.
