Abstract
We apply the theory of generalized polynomial identities with automorphisms and skew derivations to prove the following theorem: Let A be a prime ring with the extended centroid C and with two-sided Martindale quotient ring Q, R a nonzero right ideal of A and δ a nonzero σ-derivation of A, where σ is an epimorphism of A. For x, y∈ A, we set [x, y] = xy- yx. If [[…[[δ(xn0),xn1],xn2],…],xnk]=0 for all x∈ R, where n0, n1, … , nk are fixed positive integers, then one of the following conditions holds: (1) A is commutative; (2) C≅ GF(2) , the Galois field of two elements; (3) there exist b∈ Q and λ∈ C such that δ(x) = σ(x) b- bx for all x∈ A, (b- λ) R= 0 and σ(R) = 0. The analogous result for left ideals is also obtained. Our theorems are natural generalizations of the well-known results for derivations obtained by Lanski (Proc Am Math Soc 125:339–345, 1997) and Lee (Can Math Bull 38:445–449, 1995).
| Original language | English |
|---|---|
| Pages (from-to) | 833-852 |
| Number of pages | 20 |
| Journal | Monatshefte fur Mathematik |
| Volume | 180 |
| Issue number | 4 |
| DOIs | |
| Publication status | Published - 2016 Aug 1 |
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All Science Journal Classification (ASJC) codes
- Mathematics(all)
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An Engel condition with skew derivations for one-sided ideals. / Liu, Cheng Kai.
In: Monatshefte fur Mathematik, Vol. 180, No. 4, 01.08.2016, p. 833-852.Research output: Contribution to journal › Article
TY - JOUR
T1 - An Engel condition with skew derivations for one-sided ideals
AU - Liu, Cheng Kai
PY - 2016/8/1
Y1 - 2016/8/1
N2 - We apply the theory of generalized polynomial identities with automorphisms and skew derivations to prove the following theorem: Let A be a prime ring with the extended centroid C and with two-sided Martindale quotient ring Q, R a nonzero right ideal of A and δ a nonzero σ-derivation of A, where σ is an epimorphism of A. For x, y∈ A, we set [x, y] = xy- yx. If [[…[[δ(xn0),xn1],xn2],…],xnk]=0 for all x∈ R, where n0, n1, … , nk are fixed positive integers, then one of the following conditions holds: (1) A is commutative; (2) C≅ GF(2) , the Galois field of two elements; (3) there exist b∈ Q and λ∈ C such that δ(x) = σ(x) b- bx for all x∈ A, (b- λ) R= 0 and σ(R) = 0. The analogous result for left ideals is also obtained. Our theorems are natural generalizations of the well-known results for derivations obtained by Lanski (Proc Am Math Soc 125:339–345, 1997) and Lee (Can Math Bull 38:445–449, 1995).
AB - We apply the theory of generalized polynomial identities with automorphisms and skew derivations to prove the following theorem: Let A be a prime ring with the extended centroid C and with two-sided Martindale quotient ring Q, R a nonzero right ideal of A and δ a nonzero σ-derivation of A, where σ is an epimorphism of A. For x, y∈ A, we set [x, y] = xy- yx. If [[…[[δ(xn0),xn1],xn2],…],xnk]=0 for all x∈ R, where n0, n1, … , nk are fixed positive integers, then one of the following conditions holds: (1) A is commutative; (2) C≅ GF(2) , the Galois field of two elements; (3) there exist b∈ Q and λ∈ C such that δ(x) = σ(x) b- bx for all x∈ A, (b- λ) R= 0 and σ(R) = 0. The analogous result for left ideals is also obtained. Our theorems are natural generalizations of the well-known results for derivations obtained by Lanski (Proc Am Math Soc 125:339–345, 1997) and Lee (Can Math Bull 38:445–449, 1995).
UR - http://www.scopus.com/inward/record.url?scp=84933566472&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=84933566472&partnerID=8YFLogxK
U2 - 10.1007/s00605-015-0780-1
DO - 10.1007/s00605-015-0780-1
M3 - Article
AN - SCOPUS:84933566472
VL - 180
SP - 833
EP - 852
JO - Monatshefte fur Mathematik
JF - Monatshefte fur Mathematik
SN - 0026-9255
IS - 4
ER -