On 0-simple semirings, semigroup semirings and two kinds of division semirings.
On 14-dimensional quadratic forms in , 8-dimensional forms in , and the common value property.
On automorphisms of quaternion orders.
On biregular and regular rings
On -commuting graph of matrix algebra.
On elementary equivalence, isomorphism and isogeny
Motivated by recent work of Florian Pop, we study the connections between three notions of equivalence of function fields: isomorphism, elementary equivalence, and the condition that each of a pair of fields can be embedded in the other, which we call isogeny. Some of our results are purely geometric: we give an isogeny classification of Severi-Brauer varieties and quadric surfaces. These results are applied to deduce new instances of “elementary equivalence implies isomorphism”: for all genus zero...
On finite homomorphic images of the multiplicative group of a division algebra.
On free subgroups of units in quaternion algebras
It is well known that for the ring H(ℤ) of integral quaternions the unit group U(H(ℤ) is finite. On the other hand, for the rational quaternion algebra H(ℚ), its unit group is infinite and even contains a nontrivial free subgroup. In this note (see Theorem 1.5 and Corollary 2.6) we find all intermediate rings ℤ ⊂ A ⊆ ℚ such that the group of units U(H(A)) of quaternions over A contains a nontrivial free subgroup. In each case we indicate such a subgroup explicitly. We do our best to keep the arguments...
On free subgroups of units in quaternion algebras II
Let A ⊆ ℚ be any subring. We extend our earlier results on unit groups of the standard quaternion algebra H(A) to units of certain rings of generalized quaternions H(A,a,b) = ((-a,-b)/A), where a,b ∈ A. Next we show that there is an algebra embedding of the ring H(A,a,b) into the algebra of standard Cayley numbers over A. Using this embedding we answer a question asked in the first part of this paper.
On K2 of Finite Dimensional Division Algebras Over Arithmetical Fields.
On ordered division rings
Prestel introduced a generalization of the notion of an ordering of a field, which is called a semiordering. Prestel’s axioms for a semiordered field differ from the usual (Artin-Schreier) postulates in requiring only the closedness of the domain of positivity under for nonzero , instead of requiring that positive elements have a positive product. In this work, this type of ordering is studied in the case of a division ring. It is shown that it actually behaves the same as in the commutative...
On Quasi-Injectivity and von Neumann Regularity.
On quotient rings
On the Brauer monoid of .
On the existence of projective and quasi-projective covers
On the Group of Norm 1 Elements in a Division Algebra.
On the -torsion subgroup of the Brauer group of a number field
Given a number field Galois over the rational field , and a positive integer prime to the class number of , there exists an abelian extension (of exponent ) such that the -torsion subgroup of the Brauer group of is equal to the relative Brauer group of .
On the Represantation Theory of Artinian Rings and Artins Problems on Division Ring Extensions
On the roots of polynomials over division algebras.
On the zeros of a quaternionic polynomial: An extension of the Eneström-Kakeya theorem
We present some results on the location of zeros of regular polynomials of a quaternionic variable. We derive new bounds of Eneström-Kakeya type for the zeros of these polynomials by virtue of a maximum modulus theorem and the structure of the zero sets of a regular product established in the newly developed theory of regular functions and polynomials of a quaternionic variable. Our results extend some classical results from complex to the quaternionic setting as well.