On the integral ideals of \(R[X]\) when \(R\) is a special principal ideal ring. (English) Zbl 1451.13028

Summary: 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 \(\overline{I}\).


13B25 Polynomials over commutative rings
13F20 Polynomial rings and ideals; rings of integer-valued polynomials
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