Radicals in differential polynomial rings

2020 ◽  
pp. 1-17
Author(s):  
Jongwook Baeck ◽  
Nam Kyun Kim ◽  
Yang Lee
Keyword(s):  
Differential Polynomial ◽  
Prime Radical ◽  
Polynomial Rings

In this paper, we present new characterizations of several radicals of differential polynomial rings, including the Levitzki radical, strongly prime radical, and uniformly strongly prime radical in terms of the related [Formula: see text]-radical.

Derivations and bounded nilpotence index

2015 ◽  
Vol 25 (03) ◽  
pp. 433-438 ◽  
Author(s):  
Pace P. Nielsen ◽  
Michał Ziembowski
Keyword(s):  
Polynomial Ring ◽  
Differential Polynomial ◽  
Prime Radical ◽  
Polynomial Rings ◽  
Locally Nilpotent ◽  
Differential Polynomial Ring ◽  
Index Of Nilpotence ◽  
Nil Ring ◽  
Strong Barrier

We construct a nil ring R which has bounded index of nilpotence 2, is Wedderburn radical, and is commutative, and which also has a derivation δ for which the differential polynomial ring R[x;δ] is not even prime radical. This example gives a strong barrier to lifting certain radical properties from rings to differential polynomial rings. It also demarcates the strength of recent results about locally nilpotent PI rings.

scholarly journals Differential polynomial rings over rings satisfying a polynomial identity

2015 ◽  
Vol 423 ◽  
pp. 28-36 ◽  
Author(s):  
Jason P. Bell ◽  
Blake W. Madill ◽  
Forte Shinko
Keyword(s):  
Polynomial Identity ◽  
Differential Polynomial ◽  
Polynomial Rings

scholarly journals MAXIMAL COMMUTATIVE SUBRINGS AND SIMPLICITY OF ORE EXTENSIONS

2013 ◽  
Vol 12 (04) ◽  
pp. 1250192 ◽  
Author(s):  
JOHAN ÖINERT ◽  
JOHAN RICHTER ◽  
SERGEI D. SILVESTROV
Keyword(s):  
Polynomial Ring ◽  
Sufficient Conditions ◽  
Differential Polynomial ◽  
Nonzero Ideal ◽  
Polynomial Rings ◽  
Ore Extension ◽  
Differential Polynomial Ring ◽  
Ore Extensions ◽  
Necessary And Sufficient ◽  
Commutative Subring

The aim of this paper is to describe necessary and sufficient conditions for simplicity of Ore extension rings, with an emphasis on differential polynomial rings. We show that a differential polynomial ring, R[x; id R, δ], is simple if and only if its center is a field and R is δ-simple. When R is commutative we note that the centralizer of R in R[x; σ, δ] is a maximal commutative subring containing R and, in the case when σ = id R, we show that it intersects every nonzero ideal of R[x; id R, δ] nontrivially. Using this we show that if R is δ-simple and maximal commutative in R[x; id R, δ], then R[x; id R, δ] is simple. We also show that under some conditions on R the converse holds.

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