Associated Primes of Powers of Monomial Ideals: A Survey
Issue:
Volume 6, Issue 1, February 2020
Pages:
1-13
Received:
21 October 2019
Accepted:
12 November 2019
Published:
30 December 2019
Abstract: Let R be a commutative Noetherian ring and I be an ideal of R. We say that I satisfies the persistence property if AssR(R/Ik) ⊆ AssR(R/Ik+1) for all positive integers k, where AssR(R/I ) denotes the set of associated prime ideals of I. In addition, an ideal I has the strong persistence property if (Ik+1: RI) = Ik for all positive integers k. Also, an ideal I is called normally torsion-free if AssR(R/Ik) ⊆ AssR(R/I) for all positive integers k. In this paper, we collect the latest results in associated primes of powers of monomial ideals in three concepts, i.e., the persistence property, strong persistence property, and normally torsion-freeness. Also, we present some classes of monomial ideals such that are none of edge ideals, cover ideals, and polymatroidal ideals, but satisfy the persistence property and strong persistence property. In particular, we study the Alexander dual of path ideals of unrooted starlike trees. Furthermore, we probe the normally torsion-freeness of the Alexander dual of some path ideals which are related to banana trees. We close this paper with exploring the normally torsion-freeness under some monomial operations.
Abstract: Let R be a commutative Noetherian ring and I be an ideal of R. We say that I satisfies the persistence property if AssR(R/Ik) ⊆ AssR(R/Ik+1) for all positive integers k, where AssR(R/I ) denotes the set of associated prime ideals of I. In addition, an ideal I has the strong persistence property if (Ik+1: RI) = Ik for all positive integers k. Also, ...
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Volume Integral Mean of Holomorphic Function on Polydisc
Lijuan Xu,
Hua Liu,
Juan Chen,
Xiaoli Bian
Issue:
Volume 6, Issue 1, February 2020
Pages:
14-18
Received:
17 December 2019
Accepted:
4 January 2020
Published:
13 January 2020
Abstract: Let f be an analytic function in the Hardy space on the polydisc P2. In this article we discuss the area integral means Mp (f, r) of f on the polydisc P2 with radius r, and its weighted volume means Mp,α (f, r) with to the weight (1-|z1|2)a×(1-|z2|2)a. We prove that both Mp (f, r) and Mp,α (f, r) are strictly increasing in r unless f is a constant. In contrast to the classical case, we also give a example to show that log Mp,α (f, r) is not always convex with respect to log r, although that we still prove that log Mp (f, r) is logarithmically convex.
Abstract: Let f be an analytic function in the Hardy space on the polydisc P2. In this article we discuss the area integral means Mp (f, r) of f on the polydisc P2 with radius r, and its weighted volume means Mp,α (f, r) with to the weight (1-|z1|2)a×(1-|z2|2)a. We prove that both Mp (f, r) and Mp,α (f, r) are strictly increasing in r unless f is a constant....
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