I was asked to prove that, given $A$ a ring, it is equivalent to say that "every submodule of a projective module is projective" and that "every submodule of an injective module is injective". However, that is not true for $A=\mathbb{Z}$, as the first statement holds ($\mathbb{Z}$ is hereditary) but not the second.
Is there some characterization for the rings that satisfy this second statement? That is, for which "every submodule of an injective module is injective.
Every module is a submodule of an injective module. So a ring with your property would have all its modules injective. That does exist: fields have this property. I'm not sure what the exact characterization is.