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Monomial sequences of linear type. (English) Zbl 1182.13005

Summary: Let \(R\) be a noetherian commutative ring, \(\langle a_{1}, \cdots , a_n\rangle \) a sequence of elements of \(R, I=(a_1, \cdots , a_n)\) the ideal generated by the elements \(a_i\) and \(I_i=(a_1, \cdots , a_i)\), \(i=0, 1, \cdots , n\), the ideal generated by the first \(i\) elements of the sequence. A c-sequence is a sequence \(\langle a_{1}, \cdots , a_n\rangle \) which satisfies the condition \([I_{i - 1}I^k : a_i]\cap I^k=I_{i-1}I^{k- 1}\) for every \(i\in {1, \cdots , n}\) and every \(k\geq 1\). It generates an ideal of linear type. We characterize c-sequences in terms of the corresponding sequences in the Rees algebra of the ideal generated by the elements of the sequence. We then characterize monomial c-sequences of three terms.

MSC:

13A30 Associated graded rings of ideals (Rees ring, form ring), analytic spread and related topics
13B25 Polynomials over commutative rings
13A15 Ideals and multiplicative ideal theory in commutative rings
13C13 Other special types of modules and ideals in commutative rings
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Full Text: Euclid

References:

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