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Big polynomial rings and Stillman’s conjecture

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1 Published in 2019 at "Inventiones mathematicae"

DOI: 10.1007/s00222-019-00889-y

Abstract: Ananyan-Hochster's recent proof of Stillman's conjecture reveals a key principle: if $f_1, \dots, f_r$ are elements of a polynomial ring such that no linear combination has small strength then $f_1, \dots, f_r$ behave approximately like… read more here.

Keywords: big polynomial; rings stillman; stillman conjecture; polynomial rings ... See more keywords

A note on retracts of polynomial rings in three variables

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1 Published in 2019 at "Journal of Algebra"

DOI: 10.1016/j.jalgebra.2019.05.040

Abstract: In Costa's paper published in 1977, he asks us whether every retract of $k^{[n]}$ is also the polynomial ring or not, where $k$ is a field. In this paper, we give an affirmative answer in… read more here.

Keywords: three variables; retracts polynomial; polynomial rings; rings three ... See more keywords

Iterated differential polynomial rings over locally nilpotent rings

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0 Published in 2020 at "Communications in Algebra"

DOI: 10.1080/00927872.2020.1797075

Abstract: Abstract We study iterated differential polynomial rings over a locally nilpotent ring and show that a large class of such rings are Behrens radical. This extends results of Chebotar and Chen et al. read more here.

Keywords: rings locally; nilpotent rings; differential polynomial; polynomial rings ... See more keywords

On zero-divisor graphs of skew polynomial rings over non-commutative rings

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0 Published in 2017 at "Journal of Algebra and Its Applications"

DOI: 10.1142/s0219498817500566

Abstract: In this paper, we continue to study zero-divisor properties of skew polynomial rings R[x; α,δ], where R is an associative ring equipped with an endomorphism α and an α-derivation δ. For an associative ring R,… read more here.

Keywords: divisor graphs; graphs skew; zero divisor; polynomial rings ... See more keywords

Quasi-duo differential polynomial rings

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1 Published in 2017 at "Journal of Algebra and Its Applications"

DOI: 10.1142/s021949881850072x

Abstract: In this article we give a characterization of left (right) quasi-duo differential polynomial rings. We provide non-trivial examples of such rings and give a complete description of the maximal idea ... read more here.

Keywords: quasi duo; duo differential; differential polynomial; polynomial rings ... See more keywords

Chevalley groups of polynomial rings over Dedekind domains

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0 Published in 2019 at "Journal of Group Theory"

DOI: 10.1515/jgth-2019-0100

Abstract: Abstract Let R be a Dedekind domain and G a split reductive group, i.e. a Chevalley–Demazure group scheme, of rank ≥2{\geq 2}. We prove that G⁢(R⁢[x1,…,xn])=G⁢(R)⁢E⁢(R⁢[x1,…,xn]) for any⁢n≥1.G(R[x_{1},\ldots,x_{n}])=G(R)E(R[x_{1},\ldots,x_{n}])\quad\text{for any}\ n% \geq 1. In particular, this extends… read more here.

Keywords: groups polynomial; polynomial rings; group; chevalley groups ... See more keywords