I am doing the following problem:

Let $R$ be a ring, $x\in R$ a central non-unit non-zerodivisor. If $A\neq 0$ is an $R/xR$ module with $id_{R/xR}A$ finite, then $$id_R(A)=1+id_{R/xR}A$$ where $id_{R}(id_{R/xR})$ means injective dimension as a module over $R(R/xR)$

This is Exercise 4.3.3 of Weibel's Introduction to homological algebra. I am trying to mimic the proof of the corresponding theorem for projective dimension (theorem 4.3.3 in the book), which uses induction on $n=pd_{R/x}A$. The problem is that I cannot prove the base case, which says that:

If $A$ is an injective $R/xR$ module, then $id_A=1$.

I can prove that $A$ is not an injective $R$ module, so $id_RA\geq1$, but I cannot prove the other inequality. Can anyone give some hints to me? Thanks in Advance.



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