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 A note on extensions of Baer and P. P. -rings

1974 ◽  
Vol 18 (4) ◽  
pp. 470-473 ◽  
Author(s):  
Efraim P. Armendariz
Keyword(s):  
Left Ideal ◽  
Polynomial Identity ◽  
Polynomial Rings ◽  
Nilpotent Elements ◽  
Baer Ring ◽  
Baer Rings ◽  
Left Annihilator

Baer rings are rings in which the left (right) annihilator of each subset is generated by an idempotent [6]. Closely related to Baer rings are left P.P.-rings; these are rings in which each principal left ideal is projective, or equivalently, rings in which the left annihilator of each element is generated by an idempotent. Both Baer and P.P.-rings have been extensively studied (e.g. [2], [1], [3], [7]) and it is known that both of these properties are not stable relative to the formation of polynomial rings [5]. However we will show that if a ring R has no nonzero nilpotent elements then R[X] is a Baer or P.P.-ring if and only if R is a Baer or P.P.-ring. This generalizes a result of S. Jøndrup [5] who proved stability for commutative P.P.-rings via localizations – a technique which is, of course, not available to us. We also consider the converse to the well-known result that the center of a Baer ring is a Baer ring [6] and show that if R has no nonzero nilpotent elements, satisfies a polynomial identity and has a Baer ring as center, then R must be a Baer ring. We include examples to illustrate that all the hypotheses are needed.

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.

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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