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Let $A$ be a ring with identity, $\sigma$ a ring endomorphism of $A$ thatmaps the identity to itself, $\delta$ a $\sigma$-derivation of $A$, andconsider the skew-polynomial ring $A[X;\sigma,\delta]$. When $A$ is a finitefield, a Galois ring, or a general ring, some fairly recent literature used$A[X;\sigma,\delta]$ to construct new interesting codes (e.g. skew-cyclic andskew-constacyclic codes) that generalize their classical counterparts overfinite fields (e.g. cyclic and constacyclic linear codes). This paper presentsresults concerning {\it principal} $(f, \sigma, \delta)$-codes over a ring $A$,where $f\in A[X;\sigma,\delta]$ is monic. We provide recursive formulas thatcompute the entries of both a generating matrix and a control matrix of such acode $\mathcal{C}$. When $A$ is a finite commutative ring with identity and$\sigma$ is a ring automorphism of $A$, we also give recursive formulas for theentries of a parity-check matrix of $\mathcal{C}$. Also in this case, with$\delta=0$, we give a generating matrix of the dual $\mathcal{C}^\perp$,present a characterization of principal $\sigma$-codes whose duals are alsoprincipal $\sigma$-codes, and deduce a characterization of self-dual principal$\sigma$-codes. Some corollaries concerning principal $\sigma$-constacycliccodes are also given, and some highlighting examples are provided.

Characterizations and properties of principal $(f, \sigma, \delta)$-codes over rings