An Extension of a Commutativity Theorem of M. Uchiyama

In this paper we generalize some well-known commutativity theorems for associative rings as follows: LetRbe a lefts-unital ring. If there exist nonnegative integersm>1,k≥0, andn≥0such that for anyx,yinR,[xky−xnym,x]=0, thenRis commutative.

Download Full-text

THE LOCAL COMMUTATIVITY THEOREM

Form Factors in Completely Integrable Models of Quantum Field Theory - Advanced Series in Mathematical Physics ◽

10.1142/9789812798312_0003 ◽

1992 ◽

pp. 17-28

Keyword(s):

Commutativity Theorem ◽

Local Commutativity

Download Full-text

A Commutativity theorem for semiprime rings

Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics ◽

10.1017/s1446788700021881 ◽

1980 ◽

Vol 30 (1) ◽

pp. 33-36

Author(s):

Vishnu Gupta

Keyword(s):

Positive Integer ◽

Semiprime Ring ◽

Commutativity Theorem ◽

Semiprime Rings

AbstractIt is shown that if R is a semiprime ring with 1 satisfying the property that, for each x, y ∈ R, there exists a positive integer n depending on x and y such that (xy)k − xkyk is central for k = n,n+1, n+2, then R is commutative, thus generalizing a result of Kaya.

Download Full-text

A generalized commutativity theorem for pk-quasihyponormal operators

Filomat ◽

10.2298/fil0702077d ◽

2007 ◽

Vol 21 (2) ◽

pp. 77-83

Author(s):

B.P. Duggal

Keyword(s):

Hilbert Space ◽

Generalized Derivation ◽

Elementary Operator ◽

Commutativity Theorem ◽

Hilbert Space Operators

For Hilbert space operators A and B, let ?AB denote the generalized derivation ?AB(X) = AX - XB and let /\AB denote the elementary operator rAB(X) = AXB-X. If A is a pk-quasihyponormal operator, A ? pk - QH, and B*is an either p-hyponormal or injective dominant or injective pk - QH operator (resp., B*is an either p-hyponormal or dominant or pk - QH operator), then ?AB(X) = 0 =? SA*B*(X) = 0 (resp., rAB(X) = 0 =? rA*B*(X) = 0). .

Download Full-text

A Commutativity Theorem for Rings with Involution

Canadian Journal of Mathematics ◽

10.4153/cjm-1978-094-x ◽

1978 ◽

Vol 30 (6) ◽

pp. 1121-1143

Author(s):

M. Chacron

Keyword(s):

Prime Ring ◽

Associative Ring ◽

Commutativity Theorem ◽

Period 2 ◽

Rings With Involution ◽

Symmetric Element

A ring with involution R is an associative ring endowed with an antiautomorphism * of period 2. One of the first commutativity results for rings with * is a theorem of S. Montgomery asserting that if R is a prime ring, in which every symmetric element s = s* is of the form s — sn(s) (n(s) ≧ 2), then either R is commutative or R is the 2 X 2 matrices over a field, which is a nice generalization of a well-known theorem of N. Jacobson on rings all of whose elements x = xn(x).

Download Full-text