2017/08/16 by Louiza Fouli, Fouli, Louiza, Bruce Olberding +1
Computer Science · Mathematics · #13A15 #13A30 #13B22 #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Polynomial and algebraic computation
paper · pdf · doi:10.48550/arxiv.1708.04770
openalex publication_date 2017/08/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let (R, mathfrakm) be a local Noetherian ring with residue field k.\nWhile much is known about the generating sets of reductions of ideals of R if\nk is infinite, the case in which k is finite is less well understood. We\ninvestigate the existence (or lack thereof) of proper reductions of an ideal of\nR and the number of generators needed for a reduction in the case k is a\nfinite field. When R is one-dimensional, we give a formula for the smallest\ninteger n for which every ideal has an n-generated reduction. It follows\nthat in a one-dimensional local Noetherian ring every ideal has a principal\nreduction if and only if the number of maximal ideals in the normalization of\nthe reduced quotient of R is at most |k|. In higher dimensions, we show\nthat for any positive integer, there exists an ideal of R that does not have\nan n-generated reduction and that if n \≥ \dim R this ideal can be chosen\nto be mathfrakm-primary. In the case where R is a two-dimensional\nregular local ring, we construct an example of an integrally closed\n mathfrakm-primary ideal that does not have a 2-generated reduction and\nthus answer in the negative a question raised by Heinzer and Shannon.\n