Abstract
Let
A
be a ring with identity,
σ
a ring endomorphism of
A
that maps the identity to itself,
δ
a
σ
-derivation of
A
, and consider the skew-polynomial ring
A
X
;
σ
,
δ
. When
A
is a finite field, a Galois ring, or a general ring, some fairly recent literature used
A
X
;
σ
,
δ
to construct new interesting codes (e.g., skew-cyclic and skew-constacyclic codes) that generalize their classical counterparts over finite fields (e.g., cyclic and constacyclic linear codes). This paper presents results concerning monic principal skew codes, called herein monic principal
f
,
σ
,
δ
-codes, where
f
∈
A
X
;
σ
,
δ
is monic. We provide recursive formulas that compute the entries of both a generator matrix and a control matrix of such a code
C
. When
A
is a finite commutative ring and
σ
is a ring automorphism of
A
, we also give recursive formulas for the entries of a parity-check matrix of
C
. Also, in this case, with
δ
=
0
, we present a characterization of monic principal
σ
-codes whose dual codes are also monic principal
σ
-codes, and we deduce a characterization of self-dual monic principal
σ
-codes. Some corollaries concerning monic principal
σ
-constacyclic codes are also given, and a good number of highlighting examples is provided.