Geodesic
Characterization of irreducible polynomials over a special principal ideal ring
Mathematica Bohemica, Tome 148 (2023) no. 4, pp. 501-506
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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 $e$.
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 $e$.
DOI : 10.21136/MB.2022.0187-21
Classification : 13B25, 13F20
Keywords: polynomial; irreducibility; commutative principal ideal ring 
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Boudine, Brahim. Characterization of irreducible polynomials over a special principal ideal ring. Mathematica Bohemica, Tome 148 (2023) no. 4, pp. 501-506. doi: 10.21136/MB.2022.0187-21

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