Suppose that $A, B$ are commutative rings with identity, and we have an injective ring homomorphism $A \hookrightarrow B$.
I would like to know wether or not the following proposition is true: If $B$ is free as an $A$-module, then the preimage of any maximal ideal of $B$ is maximal in $A$.
Motivation: I want to show that any maximal ideal of a group ring $R[G]$ ($G$ is an arbitrary group) over a local ring $R$ contains $1\cdot \mathfrak m_R$ where $\mathfrak m_R$ is the unique maximal ideal of $R$.