Abstract
Let K be a field and R = K[x
1,…, x
n
] be the polynomial ring in the variables x
1,…, x
n
. In this paper we prove that when 𝔄 = {𝔭1,…, 𝔭
m
} and are two arbitrary sets of monomial prime ideals of R, then there exist monomial ideals I and J of R such that I ⊆ J, Ass∞(I) = 𝔄 ∪ 𝔅, Ass
R
(R/J) = 𝔅, and Ass
R
(J/I) = 𝔄 \ 𝔅, where Ass∞(I) is the stable set of associated prime ideals of I. Also we show that when 𝔭1,…, 𝔭
m
are nonzero monomial prime ideals of R generated by disjoint nonempty subsets of {x
1,…, x
n
}, then there exists a square-free monomial ideal I such that Ass
R
(R/I
k
) = Ass∞(I) = {𝔭1,…, 𝔭
m
} for all k ≥ 1.
Key Words:
ACKNOWLEDGMENTS
The authors are deeply grateful to the referee for careful reading of the manuscript and helpful suggestions.
Notes
Communicated by J. Alev.