Displaying similar documents to “Natural homomorphisms of Witt rings of orders in algebraic number fields. II”

Another look at real quadratic fields of relative class number 1

Debopam Chakraborty, Anupam Saikia (2014)

Acta Arithmetica

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The relative class number H d ( f ) of a real quadratic field K = ℚ (√m) of discriminant d is defined to be the ratio of the class numbers of f and K , where K denotes the ring of integers of K and f is the order of conductor f given by ℤ + f K . R. Mollin has shown recently that almost all real quadratic fields have relative class number 1 for some conductor. In this paper we give a characterization of real quadratic fields with relative class number 1 through an elementary approach considering the...

Automorphisms of completely primary finite rings of characteristic p

Chiteng'a John Chikunji (2008)

Colloquium Mathematicae

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A completely primary ring is a ring R with identity 1 ≠ 0 whose subset of zero-divisors forms the unique maximal ideal . We determine the structure of the group of automorphisms Aut(R) of a completely primary finite ring R of characteristic p, such that if is the Jacobson radical of R, then ³ = (0), ² ≠ (0), the annihilator of coincides with ² and R / ≅ G F ( p r ) , the finite field of p r elements, for any prime p and any positive integer r.

A commutativity theorem for associative rings

Mohammad Ashraf (1995)

Archivum Mathematicum

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Let m > 1 , s ≥ 1 be fixed positive integers, and let R be a ring with unity 1 in which for every x in R there exist integers p = p ( x ) ≥ 0 , q = q ( x ) ≥ 0 , n = n ( x ) ≥ 0 , r = r ( x ) ≥ 0 such that either x p [ x n , y ] x q = x r [ x , y m ] y s or x p [ x n , y ] x q = y s [ x , y m ] x r for all y ∈ R . In the present paper it is shown that R is commutative if it satisfies the property Q ( m ) (i.e. for all x , y ∈ R , m [ x , y ] = 0 implies [ x , y ] = 0 ).

Grothendieck and Witt groups in the reduced theory of quadratic forms

Andrzej Sładek (1980)

Annales Polonici Mathematici

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Abstract. Let F be a formally real field. Denote by G(F) and G t ( F ) the Grothen-dieck group of quadratic forms over F and its torsion subgroup, respectively. In this paper we study the structure of the factor group G ( F ) / G t ( F ) . This reduced Grothendieck group is a free Abelian group. The main results of the paper describe some sets of generators for G ( F ) / G t ( F ) , which in many cases allow us to find a basis for the group. Throughout the paper we use the language of the reduced theory of quadratic forms. In the...

Derivations with power central values on Lie ideals in prime rings

Basudeb Dhara, Rajendra K. Sharma (2008)

Czechoslovak Mathematical Journal

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Let R be a prime ring of char R ≠ 2 with a nonzero derivation d and let U be its noncentral Lie ideal. If for some fixed integers n 1 ≥ 0 , n 2 ≥ 0 , n 3 ≥ 0 , ( u n 1 [ d ( u ) , u ] u n 2 ) n 3 ∈ Z ( R ) for all u ∈ U , then R satisfies S 4 , the standard identity in four variables.

On p 2 -Ranks in the Class Field Tower Problem

Christian Maire, Cam McLeman (2014)

Annales mathématiques Blaise Pascal

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Much recent progress in the 2-class field tower problem revolves around demonstrating infinite such towers for fields – in particular, quadratic fields – whose class groups have large 4-ranks. Generalizing to all primes, we use Golod-Safarevic-type inequalities to analyse the source of the p 2 -rank of the class group as a quantity of relevance in the p -class field tower problem. We also make significant partial progress toward demonstrating that all real quadratic number fields whose class...

Derivations with Engel conditions in prime and semiprime rings

Shuliang Huang (2011)

Czechoslovak Mathematical Journal

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Let R be a prime ring, I a nonzero ideal of R , d a derivation of R and m , n fixed positive integers. (i) If ( d [ x , y ] ) m = [ x , y ] n for all x , y ∈ I , then R is commutative. (ii) If Char R ≠ 2 and [ d ( x ) , d ( y ) ] m = [ x , y ] n for all x , y ∈ I , then R is commutative. Moreover, we also examine the case when R is a semiprime ring.