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Monomial ideals with tiny squares and Freiman ideals

Ibrahim Al-Ayyoub, Mehrdad Nasernejad (2021)

Czechoslovak Mathematical Journal

We provide a construction of monomial ideals in R = K [ x , y ] such that μ ( I 2 ) < μ ( I ) , where μ denotes the least number of generators. This construction generalizes the main result of S. Eliahou, J. Herzog, M. Mohammadi Saem (2018). Working in the ring R , we generalize the definition of a Freiman ideal which was introduced in J. Herzog, G. Zhu (2019) and then we give a complete characterization of such ideals. A particular case of this characterization leads to some further investigations on μ ( I k ) that generalize some results...

Multigraded modules.

Charalambous, Hara, Deno, Christa (2001)

The New York Journal of Mathematics [electronic only]

Newton and Schinzel sequences in quadratic fields

David Adam, Paul-Jean Cahen (2010)

Actes des rencontres du CIRM

We give the maximal length of a Newton or a Schinzel sequence in a quadratic extension of a global field. In the case of a number field, the maximal length of a Schinzel sequence is 1, except in seven particular cases, and the Newton sequences are also finite, except for at most finitely many cases, all real. We give the maximal length of these sequences in the special cases. We have similar results in the case of a quadratic extension of a function field 𝔽 q ( T ) , taking in account that the ring of integers...

On Bhargava rings

Mohamed Mahmoud Chems-Eddin, Omar Ouzzaouit, Ali Tamoussit (2023)

Mathematica Bohemica

Let D be an integral domain with the quotient field K , X an indeterminate over K and x an element of D . The Bhargava ring over D at x is defined to be 𝔹 x ( D ) : = { f ∈ K [ X ] : for all a ∈ D , f ( x X + a ) ∈ D [ X ] } . In fact, 𝔹 x ( D ) is a subring of the ring of integer-valued polynomials over D . In this paper, we aim to investigate the behavior of 𝔹 x ( D ) under localization. In particular, we prove that 𝔹 x ( D ) behaves well under localization at prime ideals of D , when D is a locally finite intersection of localizations. We also attempt a classification of integral domains D ...

On local derivations in the Kadison sense

Andrzej Nowicki (2001)

Colloquium Mathematicae

Let k be a field. We prove that any polynomial ring over k is a Kadison algebra if and only if k is infinite. Moreover, we present some new examples of Kadison algebras and examples of algebras which are not Kadison algebras.

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