Some results of monomial ideals on regular sequences

Bull. Korean Math. Soc. Published online January 5, 2021

Reza Naghipour and somayeh vosughian
professor, Ph.D Candidate

Abstract : Let R denote a commutative Noetherian ring, and let x := (x_1,...,x_d) be an R-regular sequence. Suppose that a denotes a monomial ideal with respect to x. The first purpose of this article is to show that a is irreducible if and only if a is a generalized-parametric ideal. Next, it is shown that, for any integer n, (x_1,...,x_d)^n is equal to intersection of P(f), where the intersection (irredundant) is taken over all monomials f = x_1^{e_1}...x_1^{e_d}
such that deg(f ) = n - 1 and P(f ) := (x_1^{e_1+1},...,x_1^{e_d+1}. The second main result of this
paper shows that if q := (x) is a prime ideal of R which is contained in the Jacobson
radical of R and R is q-adically complete, then a is a parameter ideal if and only if a is a
monomial irreducible ideal and Rad(a) = q. In addition, if a is generated by monomials
m_1, ..., m_r; then Rad(a), the radical of a, is also monomial and Rad(a) = (rad(m_1),..., rad(m_r)).