  Given a finite ring $A$ which is a free left module over a subring $R$ of
$A$, two types of $R$-bases, pseudo-self-dual bases (similar to trace
orthogonal bases) and symmetric bases, are defined which in turn are used to
define duality preserving maps from codes over $A$ to codes over $R$. Both
types of bases are generalizations of similar concepts for fields. Many
illustrative examples are given to shed light on the advantages to such
mappings as well as their abundance.
