Let \(\mathfrak {R}_2 = \mathbb {F}_{p^m} + u\mathbb {F}_{p^m}\) with \(u^2 = 0\) , where p is an odd prime and m is any positive integer. This article explores the algebraic structure of skew negacyclic codes of length \(4p^s\) over a finite field \(\mathbb {F}_{p^m}\) and a finite chain ring \(\mathfrak {R}_2\) . Based on the different possible types of factorization of \(x^{4p^s} + 1\) over \(\mathbb {F}_{p^m}\) , a classification of the algebraic structure of skew negacyclic codes and their duals of length \(4p^s\) over \(\mathbb {F}_{p^m}\) and \(\mathfrak {R}_2\) is provided. Some necessary and sufficient conditions for self-dual codes are also discussed. Moreover, we discuss the left ideals of the quotient ring \(\frac{\mathfrak {R}_2[x;\Pi ]}{\langle ( x^2 + \delta \beta x + 1)^{p^s} \rangle }\) , where \(\Pi \) is an \(\mathbb {F}_p\) -automorphism over \(\mathfrak {R}_2\) , \(\beta ^2 = 2\) and \(\delta \in \{-1,1\}\) . Additionally, we study the torsion and residue codes of these left ideals. Examples are provided to illustrate our results, where we also obtain MDS and near-MDS codes.
