Mahdavi Hezavehi, M.

International Centre for Theoretical Physics, Trieste (Italy)

International Centre for Theoretical Physics, Trieste (Italy)

AbstractAbstract

[en] Let D be a finite dimensional F-central division algebra and v a (Krull) valuation on F. Denote by V the matrix valuation on F extending v. It is shown that the following statements are equivalent: (a) v extends to a valuation on D, (b) V defines a valuation on each K

_{ρ}, where K_{ρ}is the image of K under the regular matrix representation ρ, and K is a finite dimensional F-division subalgebra with F is included in K is included in D, (c) v has a unique extension to each finite dimensional F-division subalgebra K with F is included in K is included in D. A generalization of the above result is also given in terms of matrix valuations. (author). 8 refsPrimary Subject

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Aug 1991; 6 p

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Mahdavi Hezavehi, M.

International Centre for Theoretical Physics, Trieste (Italy)

International Centre for Theoretical Physics, Trieste (Italy)

AbstractAbstract

[en] Let D be a division ring of index n over its centre F. Denote by RN(D

^{*}) and N(D^{*}) the images of D^{*}= D - {0} under the reduced norm functions of D to F, respectively. Algebraic properties of the Abelian factor group G(D) := D^{*}/RN(D^{*})D', where D' is the derived group of D^{*}, are investigated. It is shown that there is a group homormorphism φ : G(D) → H(D), where H(D) := RN(D^{*})/N(D^{*}), whose kernel is a homormophic image of the reduced Whitehead group SK_{1}(D). Using this result, it is proved that SK_{1}(D) = {1} if and only if Ω = Z(D') and φ is an isomorphism, where Ω is the subgroup of the existing n-th roots of unity in F and Z(D') is the centre of the derived group of D^{*}. (author). 8 refsPrimary Subject

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May 1995; 6 p

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Mahdavi-Hezavehi, M.

International Centre for Theoretical Physics, Trieste (Italy)

International Centre for Theoretical Physics, Trieste (Italy)

AbstractAbstract

[en] Matrix pseudo-valuations on rings are investigated. It is proved that on a field K, each matrix pseudo-valuation is uniquely determined by its restriction to K. It is also shown how matrix valuations may be obtained from matrix pseudo-valuations by a method reminiscent of the Hahn-Banach theorem. For a skew field extension F is contained in D with (D:F)

_{L}<∞, if v is a valuation of F, it is then proved that for each fixed left basis, there exist a unique matrix pseudo-valuation p on D inducing v on F. A suitable bound for p is then calculated in terms of the matrix valuation on F. (author). 15 refsPrimary Subject

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Jun 1990; 24 p

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Mahdavi-Hezavehi, M.

International Centre for Theoretical Physics, Trieste (Italy)

International Centre for Theoretical Physics, Trieste (Italy)

AbstractAbstract

[en] Matrix orderings on rings are investigated. It is shown that in the commutative case they are essentially positive cones. This is proved by reducing it to the field case; similarly one can show that on a skew field, matrix positive cones can be reduced to positive cones by using the Dieudonne determinant. Our main result shows that there is a natural bijection between the matrix positive cones on a ring R and the ordered epic R-fields. (author). 7 refs

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Aug 1990; 8 p

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Mahdavi Hezavehi, M.; Akbari Feyzaabaadi, S.

International Centre for Theoretical Physics, Trieste (Italy)

International Centre for Theoretical Physics, Trieste (Italy)

AbstractAbstract

[en] Let D be a division ring with centre Z in which for all x not = 0 and y not = 0 in D, (xyx

^{-1}y^{-1})^{p(x,y)}is an element of Z, for some integer p(x,y) ≥ 1, depending on x and y. Given a is an element of D such that a^{s}is an element of Z for some s ≥ 1, it is shown that the minimal polynomial of a over Z is of the form t^{k}- r, where (k,6) = 1. Using this result, it is proved that if p(x,y) is of the form 2^{n(x,y)}, then D is commutative. (author). 7 refsPrimary Subject

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May 1995; 4 p

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Mahdavi Hezavehi, M.; Akbari Feyzaabaadi, S.; Mehraabaadi, M.; Hajie Abolhassan, H.

International Centre for Theoretical Physics, Trieste (Italy)

International Centre for Theoretical Physics, Trieste (Italy)

AbstractAbstract

[en] Let D be a division ring with centre F and denote by D' the derived group (commutator subgroup) of D

^{*}= D - {0}. It is shown that if each element of D' is algebraic over F, then D is algebraic over F. It is also proved that each finite separable extension of F in D is of the form F(c) for some element c in the derived group D'. Using these results, it is shown that if each element of the derived group D' is of bounded degree over F, then D is finite dimensional over F. (author). 5 refsPrimary Subject

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May 1995; 5 p

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Kuku, A.O.; Mahdavi-Hezavehi, M., E-mail: mahdavih@sharif.edu

Abdus Salam International Centre for Theoretical Physics, Trieste (Italy)

Abdus Salam International Centre for Theoretical Physics, Trieste (Italy)

AbstractAbstract

[en] Let R be a local ring, with maximal ideal m, and residue class division ring R/m=D. Put A=M

_{n}(R)-n≥1, and denote by A*=GL_{n}(R) the group of units of A. Here we investigate some algebraic structure of subnormal and maximal subgroups of A^{*}. For instance, when D is of finite dimension over its center, it is shown that finitely generated subnormal subgroups of A* are central. It is also proved that maximal subgroups of A* are not finitely generated. Furthermore, assume that P is a nonabelian maximal subgroup of GL_{1}(R) such that P contains a noncentral soluble normal subgroup of finite index, it is shown that D is a crossed product division algebra. (author)Primary Subject

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Jul 2002; 8 p; Also available at: http://www.ictp.trieste.it; 8 refs

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Kiani, D.; Mahdavi-Hezavehi, M., E-mail: kiani@mehr.sharif.edu, E-mail: mahdavih@sharif.edu

Abdus Salam International Centre for Theoretical Physics, Trieste (Italy)

Abdus Salam International Centre for Theoretical Physics, Trieste (Italy)

AbstractAbstract

[en] Let D be a division ring with centre F. Assume that M is a maximal subgroup of GL

_{n}(D), n≥1 such that Z(M) is algebraic over F. Group identities on M and polynomial identities on the F-linear hull F[M] are investigated. It is shown that if F[M] is a PI-algebra, then [D:F]<∞. When D is noncommutative and F is infinite, it is also proved that if M satisfies a group identity and [M] is algebraic over F, then we have either M=K*, where K is a field and [D:F]<∞ or M is absolutely irreducible. For a finite dimensional division algebra D, assume that N is a subnormal subgroup of GL_{n}(D) and M is a maximal subgroup of N. If M satisfies a group identity, it is shown that M is abelian-by-finite. (author)Primary Subject

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Apr 2002; 10 p; Also available at: http://www.ictp.trieste.it; 14 refs

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