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

Charkani, M. E.; Boudine, B.

São Paulo J. Math. Sci. 14, No. 2, 698-702 (2020).

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 \(\overline{I}\).

MSC:

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

Keywords:

commutative rings; polynomials; principal ideal ring; degree of polynomial

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