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On real one-sided ideals in a free algebraCimprič, Jaka ...In real algebraic geometry there are several notions of the radical of an ideal ▫$I$▫. There is the vanishing radical ▫$\sqrt{I}$▫ defined as the set of all real polynomials vanishing on the real ... zero set of ▫$I$▫, and the real radical ▫$\sqrt[re]{I}$▫ defined as the smallest real ideal containing ▫$I$▫. (Neither of them is to be confused with the usual radical from commutative algebra.) By the real Nullstellensatz, ▫$\sqrt{I} = \sqrt[re]{I}$▫. This paper focuses on extensions of these to the free algebra ▫$\mathbb{R} \langle x, x^\ast \rangle$▫ of noncommutative real polynomials in ▫$x=(x_1, \dots ,x_g)$▫ and ▫$x=(x_1^\ast, \dots ,x_g^\ast)$▫. We work with a natural notion of the (noncommutative real) zero set ▫$V(I)$▫ of a left ideal ▫$I$▫ in ▫$\mathbb{R} \langle x, x^\ast \rangle$▫. The vanishing radical ▫$\sqrt{I}$▫ of ▫$I$▫ is the set of all ▫$p \in \mathbb{R} \langle x, x^\ast \rangle$▫ which vanish on ▫$V(I)$▫. The earlier paper [J. Cimprič, J.W. Helton, S. McCullough, C. Nelson, A Non-commutative Real Nullstellensatz Corresponds to a Non-commutative Real Ideal; Algorithms, Proc. Lond. Math. Soc., 106 (2013), pp. 1060-1086] gives an appropriate notion of ▫$\sqrt[re]{I}$▫ and proves ▫$\sqrt{I} = \sqrt[re]{I}$▫ when ▫$I$▫ is a finitely generated left ideal, a free ▫$\ast$▫-Nullstellensatz. However, this does not tell us for a particular ideal I whether or not ▫$I = \sqrt[re]{I}$▫, and that is the topic of this paper. We give a complete solution for monomial ideals and homogeneous principal ideals. We also present the case of principal univariate ideals with a degree two generator and find that it is very messy. We discuss an algorithm to determine if ▫$I = \sqrt[re]{I}$▫ implemented under NCAlgebra) with finite run times and provable effectiveness.Source: Journal of Pure and Applied Algebra. - ISSN 0022-4049 (Vol. 218, iss. 2, 2014, str. 269-284)Type of material - article, component partPublish date - 2014Language - englishCOBISS.SI-ID - 16793945
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source: Journal of Pure and Applied Algebra. - ISSN 0022-4049 (Vol. 218, iss. 2, 2014, str. 269-284)
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Database name | Field | Year |
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Links to authors' personal bibliographies | Links to information on researchers in the SICRIS system |
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Cimprič, Jaka | 15127 |
Helton, J. William, 1944- | ![]() |
Klep, Igor, matematik | 22353 |
McCullough, Scott | ![]() |
Nelson, Christopher | ![]() |
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