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Centers in domains with quadratic growth

Agata Smoktunowicz (2005)

Open Mathematics

Let F be a field, and let R be a finitely-generated F-algebra, which is a domain with quadratic growth. It is shown that either the center of R is a finitely-generated F-algebra or R satisfies a polynomial identity (is PI) or else R is algebraic over F. Let r ∈ R be not algebraic over F and let C be the centralizer of r. It is shown that either the quotient ring of C is a finitely-generated division algebra of Gelfand-Kirillov dimension 1 or R is PI.

Centralizers on prime and semiprime rings

Joso Vukman (1997)

Commentationes Mathematicae Universitatis Carolinae

The purpose of this paper is to investigate identities satisfied by centralizers on prime and semiprime rings. We prove the following result: Let R be a noncommutative prime ring of characteristic different from two and let S and T be left centralizers on R . Suppose that [ S ( x ) , T ( x ) ] S ( x ) + S ( x ) [ S ( x ) , T ( x ) ] = 0 is fulfilled for all x R . If S 0 ( T ...

Classification of rings satisfying some constraints on subsets

Moharram A. Khan (2007)

Archivum Mathematicum

Let R be an associative ring with identity 1 and J ( R ) the Jacobson radical of R . Suppose that m 1 is a fixed positive integer and R an m -torsion-free ring with 1 . In the present paper, it is shown that R is commutative if R satisfies both the conditions (i) [ x m , y m ] = 0 for all x , y R J ( R ) and (ii) [ x , [ x , y m ] ] = 0 , for all x , y R J ( R ) . This result is also valid if (ii) is replaced by (ii)’ [ ( y x ) m x m - x m ( x y ) m , x ] = 0 , for all x , y R N ( R ) . Our results generalize many well-known commutativity theorems (cf. [1], [2], [3], [4], [5], [6], [9], [10], [11] and [14]).

Commutativity of associative rings through a Streb's classification

Mohammad Ashraf (1997)

Archivum Mathematicum

Let m 0 , r 0 , s 0 , q 0 be fixed integers. Suppose that R is an associative ring with unity 1 in which for each x , y R there exist polynomials f ( X ) X 2 Z Z [ X ] , g ( X ) , h ( X ) X Z Z [ X ] such that { 1 - g ( y x m ) } [ x , x r y - x s f ( y x m ) x q ] { 1 - h ( y x m ) } = 0 . Then R is commutative. Further, result is extended to the case when the integral exponents in the above property depend on the choice of x and y . Finally, commutativity of one sided s-unital ring is also obtained when R satisfies some related ring properties.

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