The primary radical of special Lie algebras

Бейдар, К. И.; Pikhtilkov, S. A.

doi:10.1070/rm1994v049n01abeh002144

Cited by 4 publications

(5 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Analogues of this theorem for Lie algebras are obtained in [4,5,23,43]. In the case of groups, a similar statement also holds and can be formulated as follows.…”

Section: Radicals Of Multiplicative Subgroups Of Associative Pi-algebrasmentioning

confidence: 73%

“…qq − e qq ze * pp = 1 for some z ∈ R/P and 1 ≤ p = q, q * ≤ 4. Therefore, by Lemma 5.4, the group B P contains all elements t pq (b), t p * q (b), t pq * (b), t p * q * (b), t p (b), t q (b), t p * (b), and t q * (b) for b ∈ (IJ) 4 , where I = R/P (t pq (z) − 1)R/P is the * -ideal of the algebra R/P generated by t pq (z) − 1 and J is the ideal of the center Z(R/P ) of this algebra generated by the elements v − v 3 , where v ∈ Z(R/P ) s . Note that by the terms on the * -prime algebra R/P and Remark 1.13, the ideals J and IJ are not equal to zero.…”

Section: Radicals Of Unitary Groups Over Rings With Involutionmentioning

confidence: 99%

“…. , 4, where 1 ≤ m = n = n * ≤ 4, and z ∈ (IJ) 4 , where I = RaR is the ideal of the algebra R generated by the element a = t pq (r) − 1 and J is the ideal of the center…”

Section: Remark 416 ([22 Lemma 26]mentioning

confidence: 99%

“…). If the assembly S consists of elements of a * -ideal (IJ) 4 , then the finite set H 0 (S) is contained in the group B P and, since this group is weakly solvable, we find a number k = k(S) ≥ 1 for which H k (S) = {1} and e mm ve nn = e mm (w − w * )e mm = 0 for all v ∈ S mn,m (k) ∪ S mn,n (k), w ∈ S m (k), and 1 ≤ m = n, n * ≤ 4. In particular, we can take any element x ∈ (IJ) 4 and index i, where 1 ≤ i ≤ 4, and define the corresponding number k(X) = k i (x) ≥ 1 for the subset X = X i (x) = {±e pi (x − x * )e i * q | p, q = 1, .…”

Section: Radicals Of Unitary Groups Over Rings With Involutionmentioning

confidence: 99%

“…This is a contradiction (y = 0). Thus, e ii (x − x * )e * ii = 0 and e ii xe * ii = e ii x * e * ii for all x ∈ (IJ) 4 and i = 1, . .…”

Section: Radicals Of Unitary Groups Over Rings With Involutionmentioning

confidence: 99%

See 4 more Smart Citations

The radical RN and the weakly solvable radical of linear groups over associative rings

Golubkov

2008

J Math Sci

View full text Add to dashboard Cite

This paper is devoted to the computation of the radicals RN and RN * and the weakly solvable radical for a number of basic classical linear groups over rings, including the unitary group over a ring with involution and matrix groups normalized by elementary Chevalley groups.Proposition 1.7. The radical RN * (R) of an algebra R over a ring F is locally nilpotent.Proposition 1.8. The locally nilpotent radical L(R) of a countably generated algebra R over a ring F coincides with the radical RN * (R).An important role in our further constructions will be played by the possibility of expressing the radicals Rad, L, and RN * of a matrix algebra M n (R) over an algebra with unity R by means of the corresponding radicals of the algebra of coefficients R.Remark 1.9. For any algebra R with unity 1 over a ring F and each integer n ≥ 1 the equalities

show abstract

“…Analogues of this theorem for Lie algebras are obtained in [4,5,23,43]. In the case of groups, a similar statement also holds and can be formulated as follows.…”

Section: Radicals Of Multiplicative Subgroups Of Associative Pi-algebrasmentioning

confidence: 73%

Section: Radicals Of Unitary Groups Over Rings With Involutionmentioning

confidence: 99%

“…. , 4, where 1 ≤ m = n = n * ≤ 4, and z ∈ (IJ) 4 , where I = RaR is the ideal of the algebra R generated by the element a = t pq (r) − 1 and J is the ideal of the center…”

Section: Remark 416 ([22 Lemma 26]mentioning

confidence: 99%

Section: Radicals Of Unitary Groups Over Rings With Involutionmentioning

confidence: 99%

“…This is a contradiction (y = 0). Thus, e ii (x − x * )e * ii = 0 and e ii xe * ii = e ii x * e * ii for all x ∈ (IJ) 4 and i = 1, . .…”

Section: Radicals Of Unitary Groups Over Rings With Involutionmentioning

confidence: 99%

See 3 more Smart Citations