A Commutativity Theorem for Rings with Involution

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).

scholarly journals On n-generalized commutators and Lie ideals of rings

2021 ◽  
pp. 2250221
Author(s):  
Peter V. Danchev ◽  
Tsiu-Kwen Lee
Keyword(s):  
Positive Integer ◽  
Prime Ring ◽  
Classical Result ◽  
Associative Ring ◽  
Nonzero Ideal ◽  
Lie Ideal ◽  
Additive Subgroup ◽  
Simple Ring ◽  
Noncommutative Polynomials ◽  
Generalized Commutator

Let [Formula: see text] be an associative ring. Given a positive integer [Formula: see text], for [Formula: see text] we define [Formula: see text], the [Formula: see text]-generalized commutator of [Formula: see text]. By an [Formula: see text]-generalized Lie ideal of [Formula: see text] (at the [Formula: see text]th position with [Formula: see text]) we mean an additive subgroup [Formula: see text] of [Formula: see text] satisfying [Formula: see text] for all [Formula: see text] and all [Formula: see text], where [Formula: see text]. In the paper, we study [Formula: see text]-generalized commutators of rings and prove that if [Formula: see text] is a noncommutative prime ring and [Formula: see text], then every nonzero [Formula: see text]-generalized Lie ideal of [Formula: see text] contains a nonzero ideal. Therefore, if [Formula: see text] is a noncommutative simple ring, then [Formula: see text]. This extends a classical result due to Herstein [Generalized commutators in rings, Portugal. Math. 13 (1954) 137–139]. Some generalizations and related questions on [Formula: see text]-generalized commutators and their relationship with noncommutative polynomials are also discussed.

Lie Action of Certain Skews in *-Rings

1978 ◽  
Vol 30 (4) ◽  
pp. 700-710
Author(s):  
M. Chacron
Keyword(s):  
Associative Ring ◽  
Additive Subgroup ◽  
Period 2

A *-ring is an associative ring R with an anti-automorphism * of period 2 (involution). Call x ∈ R skew (symmetric) if x = - x* (x = x*) and let K(S) be the additive subgroup of all skews (symmetries). If [a, b] denotes the Lie product of a, b ∈ R (that is, ab — ba) and if [A, B] denotes the Lie product of the additive subgroups A and B of R (that is, the additive subgroup generated by [a, b], a and b ranging over A and B) then clearly [K, K] is an additive subgroup contained in K.

Structure of a Certain Class of Rings with Involution

1975 ◽  
Vol 27 (5) ◽  
pp. 1114-1126 ◽  
Author(s):  
M. Chacron ◽  
I. N. Herstein ◽  
S. Montgomery
Keyword(s):  
Integer Coefficients ◽  
Rings With Involution ◽  
Symmetric Element

Let R be a ring with involution *, and let Z denote the center of R. In R let S = {x ∈ R|x* = x} be the set of symmetric elements of R. We shall study rings which are conditioned in the following way: given s ∈ S, then for some integer and some polynomial p(t), with integer coefficients which depend on . What can one hope to say about such rings? Certainly all rings in which every symmetric element is nilpotent fall into this class.

Nilpotent Inner Derivations of the Skew Elements of Prime Rings With Involution

1991 ◽  
Vol 43 (5) ◽  
pp. 1045-1054 ◽  
Author(s):  
W. S. Martindale ◽  
C. Robert Miers
Keyword(s):  
Prime Ring ◽  
Central Simple Algebra ◽  
Simple Algebra ◽  
Extended Centroid ◽  
Prime Rings ◽  
Rings With Involution ◽  
Central Simple

AbstractLet R be a prime ring with invoution *, of characteristic 0, with skew elements K and extended centroid C. Let a ∈ K be such that (ad a)n =0 on K. It is shown that one of the following possibilities holds: (a) R is an order in a 4-dimensional central simple algebra, (b) there is a skew element λ in C such that , (c) * is of the first kind, n ≡ 0 or n ≡ 3 (mod 4), and . Examples are given illustrating (c).

scholarly journals On Maps of Period 2 on Prime and Semiprime Rings

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
H. E. Bell ◽  
M. N. Daif
Keyword(s):  
Generalized Derivations ◽  
Prime Rings ◽  
Period 2 ◽  
Prime And Semiprime Rings ◽  
Semiprime Rings ◽  
Rings With Involution

A mapfof the ringRinto itself is of period 2 iff2x=xfor allx∈R; involutions are much studied examples. We present some commutativity results for semiprime and prime rings with involution, and we study the existence of derivations and generalized derivations of period 2 on prime and semiprime rings.

scholarly journals A Note on Pair of Left Centralizers in Prime Ring with Involution

2021 ◽  
Vol 45 (02) ◽  
pp. 225-236
Author(s):  
MUZIBUR RAHMAN MOZUMDER ◽  
ADNAN ABBASI ◽  
NADEEM AHMAD DAR ◽  
AFTAB HUSSAIN SHAH
Keyword(s):  
Prime Ring ◽  
Prime Rings ◽  
Rings With Involution

The purpose of this paper is to study pair of left centralizers in prime rings with involution satisfying certain identities.

scholarly journals Generalized (α,β )-higher Derivations on Lie Ideals of Rings with Involution

2019 ◽  
pp. 1-7
Author(s):  
Abdul Nadim Khan
Keyword(s):  
Prime Ring ◽  
Functional Identity ◽  
Rings With Involution ◽  
Additive Mappings

In this manuscript, we investigate the behaviour of additive mappings which satisfy a functional identity associated with generalized (α, β)-higher derivations on Lie ideals of a prime ring with involution. As the consequences of our main theorem, many well known results can be deduced.

On Rings with Involution

1974 ◽  
Vol 26 (4) ◽  
pp. 794-799 ◽  
Author(s):  
I. N. Herstein
Keyword(s):  
Prime Ring ◽  
Simple Proof ◽  
Positive Definiteness ◽  
Joint Work ◽  
Original Proof ◽  
Recent Theorem ◽  
Rings With Involution

In this note we prove some results which assert that under certain conditions the involution on a prime ring must satisfy a form of positive definiteness. As a consequence of the first of our theorems we obtain a fairly short and simple proof of a recent theorem of Lanski [3]. In fact, in doing so we actually generalize his result in that we need not avoid the presence of 2-torsion. One can easily adapt Lanski's original proof, also, to cover the case in which 2-torsion is present. This result of Lanski has been greatly generalized in a joint work by Susan Montgomery and ourselves [2].

Rings with Involution in which Every Trace is Nilpotent or Regular

1974 ◽  
Vol 26 (1) ◽  
pp. 130-137 ◽  
Author(s):  
Susan Montgomery
Keyword(s):  
Division Ring ◽  
Simple Ring ◽  
Direct Sum ◽  
Rings With Involution ◽  
Symmetric Element

A theorem of Marshall Osborn [15] states that a simple ring with involution of characteristic not 2 in which every non-zero symmetric element is invertible must be a division ring or the 2 × 2 matrices over a field. This result has been generalized in several directions. IfRis semi-simple and every symmetric element (or skew, or trace) is invertible or nilpotent, thenRmust be a division ring, the 2 × 2 matrices over a field, or the direct sum of a division ring and its opposite [6; 8; 13; 16].

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