Good folks!
Having problem sorting out the solution to a question that should be relatively simple from Weibel's book of $K$-theory. Hoping you could help me. The question in particular is Exercise 2.1:
Let $I$ be a radical ideal in $R$. If $P_1 , P_2$ are finitely generated projective $R$-modules such that $P_1 / I P_1 \cong P_2 / I P_2$, show that $P_1 \cong P_2$. Hint: Modify the proof of 2.2, observing that $\text{Hom} (P,Q) \rightarrow \text{Hom} (P/I , Q/I)$ is onto.
Okay, I figured, maybe the idea here is to note that we should have $(P_1 / I P_1) \oplus (Q/IQ) \cong (P_2 / I P_2) \oplus (Q/IQ) \cong R^n/I$ for some $n \in \mathbb{N}$, and $R$-module $Q$, and then pick elements $\{ e_1 , \dots , e_{p_1} \} \in P_1$, $\{ e_1 , \dots , e_{p_2} \} \in P_2$, that maps to generator elements of $P_1/IP_1$ and $P_2/IP_2$ respectively, have these define morphisms $P_1 \rightarrow P_1 \oplus Q$ and $P_2 \rightarrow P_2 \oplus Q$, but I soon enough get uncertain as to whether or not this is even the right way to proceed.
Any and all help is appreciated.