On radical formula in modules over noncommutative rings

Öneş, Ortaç

doi:10.2298/fil2002443o

Cited by 3 publications

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References 5 publications

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“…Reference [3] is a different generalization of [1] that will not be discussed here. References [2] and [12] are earlier developments. 1.…”

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confidence: 99%

Matrix versions of real and quaternionic nullstellensatz

Cimprič¹

2022

Preprint

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Real Nullstellensatz is a classical result from Real Algebraic Geometry. It has recently been extended to quaternionic polynomials by Alon and Paran [1]. The aim of this paper is to extend their Quaternionic Nullstellensatz to matrix polynomials. We also obtain an improvement of the Real Nullstellensatz for matrix polynomials from [5] in the sense that we simplify the definition of a real left ideal. We use the methods from the proof of the matrix version of Hilbert's Nullstellensatz [6] and we obtain their extensions to a mildly non-commutative case and to the real case.Recall that an ideal I of a commutative ring R is real if for every a 1 , . . . , a k ∈ R such that a 2 1 + . . . + a 2 k ∈ I we have a 1 , . . . , a k ∈ I. The smallest real ideal that contains a given ideal J is called the real radical of J and it is usually denoted by rr √ J.1.2. Quaternionic polynomials. In [1], Theorem A was extended to quaternionic polynomials. Their results are summarized in subsections

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“…Reference [3] is a different generalization of [1] that will not be discussed here. References [2] and [12] are earlier developments. 1.…”

mentioning

confidence: 99%

Matrix versions of real and quaternionic nullstellensatz

Cimprič¹

2022

Preprint

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Matrix versions of Real and Quaternionic Nullstellensätze

Cimpric

2022

Journal of Algebra

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On Prime Ideals Related to an Ideal in a Commutative Ring

Öneş

2021

Gazi University Journal of Science Part A: Engineering and Innovation

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Prime IdealRadical Ideal Localization This article focuses on the notion of prime ideal related to an ideal of a commutative ring. Some of its characteristics are studied and the relations between ideals and prime ideals related to these ideals in a commutative ring are examined. Then some useful connections among them are obtained. Both similarities and differences between prime ideals related to an ideal and prime ideals are pointed out. Moreover a direct connection between prime ideals related to an ideal and radical ideals is obtained.

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On radical formula in modules over noncommutative rings

Cited by 3 publications

References 5 publications

Matrix versions of real and quaternionic nullstellensatz

Matrix versions of real and quaternionic nullstellensatz

Matrix versions of Real and Quaternionic Nullstellensätze

On Prime Ideals Related to an Ideal in a Commutative Ring

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