Abstract
Let R be a prime ring and δ a derivation of R. Divided powers, of ordinary differentiation d/dx form Hasse-Schmidt higher derivations of the Ore extension (skew polynomial ring) R[x; δ]. They have been used crucially but implicitly in the investigation of R[x; δ]. Our aim is to explore this notion. The following is proved among others: Let Q be the left Martindale quotient ring of R. It is shown that, is a quasi-injective (R, R)-module and that any (R,R)-bimodule endomorphism of S can be uniquely expressed in the form, where ζn ∈ CS(R), the centralizer of R in S. As an application, we also use the Ore extension R[x; δ] to deduce Kharchenko's theorem for a single derivation. These results are extended to the Ore extension R[X;D] of R by a sequence D of derivations of R.
| Original language | English |
|---|---|
| Pages (from-to) | 157-178 |
| Number of pages | 22 |
| Journal | Israel Journal of Mathematics |
| Volume | 175 |
| Issue number | 1 |
| DOIs | |
| Publication status | Published - 2010 Jan 1 |
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All Science Journal Classification (ASJC) codes
- Mathematics(all)
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Higher derivations of ore extensions. / Chuang, Chen Lian; Lee, Tsiu Kwen; Liu, Cheng Kai; Tsai, Yuan Tsung.
In: Israel Journal of Mathematics, Vol. 175, No. 1, 01.01.2010, p. 157-178.Research output: Contribution to journal › Article
TY - JOUR
T1 - Higher derivations of ore extensions
AU - Chuang, Chen Lian
AU - Lee, Tsiu Kwen
AU - Liu, Cheng Kai
AU - Tsai, Yuan Tsung
PY - 2010/1/1
Y1 - 2010/1/1
N2 - Let R be a prime ring and δ a derivation of R. Divided powers, of ordinary differentiation d/dx form Hasse-Schmidt higher derivations of the Ore extension (skew polynomial ring) R[x; δ]. They have been used crucially but implicitly in the investigation of R[x; δ]. Our aim is to explore this notion. The following is proved among others: Let Q be the left Martindale quotient ring of R. It is shown that, is a quasi-injective (R, R)-module and that any (R,R)-bimodule endomorphism of S can be uniquely expressed in the form, where ζn ∈ CS(R), the centralizer of R in S. As an application, we also use the Ore extension R[x; δ] to deduce Kharchenko's theorem for a single derivation. These results are extended to the Ore extension R[X;D] of R by a sequence D of derivations of R.
AB - Let R be a prime ring and δ a derivation of R. Divided powers, of ordinary differentiation d/dx form Hasse-Schmidt higher derivations of the Ore extension (skew polynomial ring) R[x; δ]. They have been used crucially but implicitly in the investigation of R[x; δ]. Our aim is to explore this notion. The following is proved among others: Let Q be the left Martindale quotient ring of R. It is shown that, is a quasi-injective (R, R)-module and that any (R,R)-bimodule endomorphism of S can be uniquely expressed in the form, where ζn ∈ CS(R), the centralizer of R in S. As an application, we also use the Ore extension R[x; δ] to deduce Kharchenko's theorem for a single derivation. These results are extended to the Ore extension R[X;D] of R by a sequence D of derivations of R.
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U2 - 10.1007/s11856-010-0007-z
DO - 10.1007/s11856-010-0007-z
M3 - Article
AN - SCOPUS:77949931063
VL - 175
SP - 157
EP - 178
JO - Israel Journal of Mathematics
JF - Israel Journal of Mathematics
SN - 0021-2172
IS - 1
ER -