Asymptotic behaviour of monomial ideals on regular sequences

  • Monireh Sedghi

    Azarbaidjan University of Tarbiat Moallem, Tabriz, Iran


Let RR be a commutative Noetherian ring, and let x=x1,,xd\mathbf{x}= x_1, \ldots, x_d be a regular RR-sequence contained in the Jacobson radical of RR. An ideal II of RR is said to be a monomial ideal with respect to x\mathbf{x} if it is generated by a set of monomials x1e1xdedx_1^{e_1}\ldots x_d^{e_d}. The monomial closure of II, denoted by I~\widetilde{I}, is defined to be the ideal generated by the set of all monomials mm such that mnInm^n\in I^n for some nNn\in \mathbb{N}. It is shown that the sequences AssRR/In~\mathrm{Ass}_RR/\widetilde{I^n} and AssRIn~/In\mathrm{Ass}_R\widetilde{I^n}/I^n, n=1,2,,n=1,2, \ldots, of associated prime ideals are increasing and ultimately constant for large nn. In addition, some results about the monomial ideals and their integral closures are included.

Cite this article

Monireh Sedghi, Asymptotic behaviour of monomial ideals on regular sequences. Rev. Mat. Iberoam. 22 (2006), no. 3, pp. 955–962

DOI 10.4171/RMI/479