Let $R$ be a commutative Noetherian ring and $I$ be an ideal of $R$. We say that $I$ satisfies the persistence property if $\mathrm{Ass}_R(R/I^k)\subseteq \mathrm{Ass}_R(R/I^{k+1})$ for all positive integers $k\geq 1$, which $\mathrm{Ass}_R(R/I)$ denotes the set of associated prime ideals of $I$. In this paper, we introduce a class of square-free monomial ideals in the polynomial ring $R=K[x_1,\ldots,x_n]$ over field $K$ which are associated to unrooted trees such that if $G$ is a unrooted tree and $I_t(G)$ is the ideal generated by the paths of $G$ of length $t$, then $J_t(G):=I_t(G)^\vee$, where $I^\vee$ denotes the Alexander dual of $I$, satisfies the persistence property. We also present a class of graphs such that the path ideals generated by paths of length two satisfy the persistence property. We conclude this paper by giving a criterion for normally torsion-freeness of monomial ideals.
Nasernejad, M. (2014). Asymptotic behaviour of associated primes of monomial ideals with combinatorial applications. Journal of Algebra and Related Topics, 2(1), 15-25.
MLA
M. Nasernejad. "Asymptotic behaviour of associated primes of monomial ideals with combinatorial applications". Journal of Algebra and Related Topics, 2, 1, 2014, 15-25.
HARVARD
Nasernejad, M. (2014). 'Asymptotic behaviour of associated primes of monomial ideals with combinatorial applications', Journal of Algebra and Related Topics, 2(1), pp. 15-25.
VANCOUVER
Nasernejad, M. Asymptotic behaviour of associated primes of monomial ideals with combinatorial applications. Journal of Algebra and Related Topics, 2014; 2(1): 15-25.
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