On the integral ideals of R [ X ] when R is a special principal ideal ring
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On the integral ideals of R[X] when R is a special principal ideal ring M. E. Charkani1 · B. Boudine 1
© Instituto de Matemática e Estatística da Universidade de São Paulo 2020
Abstract Let (R, 𝜋R, 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 𝜋 . 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 I ≠ 0 in k[X]. Furthermore, the lowest degree of monic polynomials in I is exactly the lowest degree of nonzero polynomials in I . Keywords Commutative rings · Polynomials · Principal ideal ring · Degree of polynomial Mathematics Subject Classification 13B25 · 13F20
1 Introduction The polynomial rings are very useful in cryptography, coding theory and geometry. The study of their ideals is important in classifying modules over polynomial rings. Especially, Snapper [13] has used the monic polynomials of lowest degree to prove that every ideal of R[X] for a completely primary ring R can be generated by xi fi where fi are monic polynomials. This result was indispensable for McDonald [10] to classify matrices over Artinian principal ideals rings. Also, Mandal [8] has investigated the minimal number of generators of an ideal I of a polynomial ring, then Nashier [11], Kumar and Singh [7] and Kumar and Mishra [6] worked to resolve the
Communicated by Sergio R. López-Permouth. * B. Boudine [email protected] M. E. Charkani [email protected] 1
Faculty of sciences Dhar El Mahraz, Fez, Morocco
13
Vol.:(0123456789)
São Paulo Journal of Mathematical Sciences
same problem. In all these works, the condition that I contains a monic polynomial was essential. In view of this, one may ask: 1. When does an ideal I of a polynomial ring contain a monic polynomial? 2. If such an ideal contains monic polynomials, what is the lowest degree of any among them? In this paper, we answer these questions when (R, 𝜋R, k, e) is a local principal ideal ring called a special principal ideal ring (SPIR) where 𝜋R is its maximal ideal, k its residual field and e the index of nilpotency of 𝜋 . For more information, see [14, p. 245, 3, 9, pp. 339–341, 5, pp. 114–115, 1, pp. 176–177, 12, 2 and 4]. We call an ideal I of R[X] containing a monic polynomial: an integral ideal, we denote the lowest degree of monic polynomials of I by dg0 (I) and the lowest degree of nonzero polynomials of I by dg(I). Furthermore, if I is a principal ideal of R[X], we call it a monic principal ideal if there is a monic polynomial p such that I = p.R[X] . Then we prove the following result: Theorem Let (R, 𝜋R, k, e)be a SPIR and I be an ideal in R [X]. Then, the following statements are equivalent: 1. 2. 3. 4.
I is an integral ideal of R[X]. I contains a primitive polynomial in R[X]. I − 𝜋R[X] ≠ �. I ≠ 0 in k[X].
In this case, we get dg0 (I) = dg(I). Furthermore, if I is principal, then it is a monic principal ideal.
2 Preliminaries
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