  In this work we generalize the concept of injective module and develop a
theory of divisibility for modules over a general ring, which provides a
general and unified framework to study Kummer-like field extensions arising
from commutative algebraic groups. With these tools we provide an effective
bound for the degree of the field extensions arising from division points of
elliptic curves, extending previous results of Javan Peykar for CM curves and
of Lombardo and the author for the non-CM case.
