(Assuming rings to be commutative with unity.) Given a local ring $(A,m_A)$, I want to know if given an injective map of $A$-modules $M\rightarrow N$, is the $A/m_A$-module map $M/m_AM \rightarrow N/m_AN$ injective.
I am confused because I know that taking quotients isn't left exact in general. But, I haven't been able to find a counterexample. In fact, assuming that $A$ is a discrete valuation ring, I have actually been able to apply structure theorem to convince myself that the injectivity is preserved.
Anyone got a hint?