Abstract
Let \((R, \pi R, k,e)\) be a commutative special principal ideal ring (SPIR), where R is its maximal ideal, k its residual field and e the index of nilpotency of \(\pi\). An ideal I of R[X] is called an integral ideal if it contains a monic polynomial. In this paper, we show that if R is a SPIR, then an ideal I in R[X] is integral if and only if \({\overline{I}} \ne {{\overline{0}}}\) in k[X]. Furthermore, the lowest degree of monic polynomials in I is exactly the lowest degree of nonzero polynomials in \({\overline{I}}\).
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Communicated by Sergio R. López-Permouth.
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Charkani, M.E., Boudine, B. On the integral ideals of R[X] when R is a special principal ideal ring. São Paulo J. Math. Sci. 14, 698–702 (2020). https://doi.org/10.1007/s40863-020-00177-1
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DOI: https://doi.org/10.1007/s40863-020-00177-1