Characterization of irreducible polynomials over a special principal ideal ring
Received December 5, 2021. Published online September 8, 2022.
Abstract: A commutative ring $R$ with unity is called a special principal ideal ring (SPIR) if it is a non integral principal ideal ring containing only one nonzero prime ideal, its length $e$ is the index of nilpotency of its maximal ideal. In this paper, we show a characterization of irreducible polynomials over a SPIR of length $2$. Then, we give a sufficient condition for a polynomial to be irreducible over a SPIR of any length $e$.
Keywords: polynomial; irreducibility; commutative principal ideal ring