On real one-sided ideals in a free algebra
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- 作者：Jakob Cimpri膷 ; J. William Helton ; Igor Klep ; Scott McCullough ; Christopher Nelson
- 关键词：Primary ; 14P10 ; 08B20 ; secondary ; 90C22 ; 16W10 ; 13J30
- 刊名：Journal of Pure and Applied Algebra
- 出版年：February, 2014
- 年：2014
- 卷：218
- 期：2
- 页码：269-284
- 全文大小：446 K
文摘
In real algebraic geometry there are several notions of the radical of an ideal . There is the vanishing radical defined as the set of all real polynomials vanishing on the real zero set of , and the real radical defined as the smallest real ideal containing . (Neither of them is to be confused with the usual radical from commutative algebra.) By the real Nullstellensatz, . This paper focuses on extensions of these to the free algebra of noncommutative real polynomials in and .
We work with a natural notion of the (noncommutative real) zero set of a left ideal in . The vanishing radical of is the set of all which vanish on . The earlier paper (Cimpri膷 et聽al. [6]) gives an appropriate notion of and proves when is a finitely generated left ideal, a free -Nullstellensatz. However, this does not tell us for a particular ideal whether or not , 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 (implemented under ) with finite run times and provable effectiveness.