A linear code of length $n$ on a ring $R$ is an $R$-submodule of the free $R$-module $R^n$. These are of interest in applied mathematics to ensure successful transmission of a message despite potential errors in communication.
We define a dual code of a code $C \subset R^n$ as $C^\perp = \{ x \in R^n | x \cdot c = 0 \forall c \in C \}$. I am wondering the following:
Question: Let $C \subset R^n$ be a linear code which is free as an $R$-module. Then is the dual code $C^\perp$ of $C$ also free?
Besides it's relevance to linear codes, this question could be of intrinsic interest in module theory.