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This note investigates two long-standing conjectures on the Krull dimension of integer-valued polynomial rings and of polynomial rings in the context of (locally) essential domains.
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《Journal of Pure and Applied Algebra》2023,227(2):107179
Let D be an integral domain and let be a torsion-free, ≤-cancellative, subtotally ordered monoid. We show that the generalized power series ring is a Krull domain if and only if D is a Krull domain and S is a Krull monoid. 相似文献
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M. Petrich 《Acta Mathematica Hungarica》2002,97(4):303-322
The relation in the title is S defined by
on an arbitrary semigroup. We investigate antisymmetry of S by means of a (minimal) family whose members can not appear as subsemigroups. Transitivity of S is characterized similarly by means of the family and homomorphic images of a certain semigroup. We study the transfer of certain properties of a monoid T and the Bruck semigroup B(T,) over T. The paper concludes with a consideration of certain properties of the relation S on inverse semigroups. 相似文献
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Thierry Levasseur 《Proceedings of the American Mathematical Society》2002,130(12):3519-3523
Let be a complex semisimple Lie algebra and be its enveloping algebra. We deduce from the work of R. Bezrukavnikov, A. Braverman and L. Positselskii that the Krull-Gabriel-Rentschler dimension of is equal to the dimension of a Borel subalgebra of .
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Robert Gilmer William Heinzer Moshe Roitman 《Proceedings of the American Mathematical Society》1999,127(11):3141-3151
Suppose is a maximal ideal of a commutative integral domain and that some power of is finitely generated. We show that is finitely generated in each of the following cases: (i) is of height one, (ii) is integrally closed and , (iii) is a monoid domain over a field , where is a cancellative torsion-free monoid such that , and is the maximal ideal . We extend the above results to ideals of a reduced ring such that is Noetherian. We prove that a reduced ring is Noetherian if each prime ideal of has a power that is finitely generated. For each with , we establish existence of a -dimensional integral domain having a nonfinitely generated maximal ideal of height such that is -generated.
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Bruhat-Chevalley Order in Reductive Monoids 总被引:1,自引:1,他引:0
Mohan S. Putcha 《Journal of Algebraic Combinatorics》2004,20(1):33-53
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Using actions of free monoids and free associative algebras, we establish some Schreier-type formulas involving ranks of actions and ranks of subactions in free actions or Grassmann-type relations for the ranks of intersections of subactions of free actions. The coset action of the free group is used to establish a generalization of the Schreier formula in the case of subgroups of infinite index. We also study and apply large modules over free associative and free group algebras. 相似文献
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