A note on the Alexander dual of path ideals of rooted trees

ABSTRACT

An ideal I in a commutative Noetherian ring S is called normally torsion-free if ${Ass}_S(S∕I^k)\subseteq {Ass}_S(S∕I)$ for all positive integers k, where Ass_S(S∕I) denotes the set of associated prime ideals of I. In this note, we show that the Alexander dual of path ideals generated by all paths of length 2 in rooted trees are normally torsion-free.

KEYWORDS:

• Associated primes
• normally torsion-free ideals
• rooted trees

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 13B25
• 13F20
• 05C05

Acknowledgements

The authors are deeply grateful to the referee for careful reading of the manuscript.