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In algebra, division rings or skew fields are considered one of the basic building blocks in ring theory. So, a careful study of the properties of these rings is essential for the development of ring theory. All division rings, according to whether they are finite dimensional or infinite dimensional (as vector spaces) over their centers, are broadly classified into two types of centrally finite division rings and centrally infinite division rings.Herstein's conjecture about multiplicative commutators in these rings, is one of the unsolved problems that Herstein has proved it in the case when division ring is centrally finite, or the center of division ring is uncountable, but the... 

In this thesis we investigate identities on maximal subgroups of developed by D. Kiani and M. Mahdavi-Hezavehi . Let be a division ring with centre and a maximal subgroup of ( ) . Several group identities on M and polynomial identities on the F-linear hull where is algebraic over F are studied. We show that if is a PI-algebra, then . When is non-commutative and is infinite, we show that if satisfies a group identity and is algebraic over , then we have either where is a field and , or is absolutely irreducible. Finally for a finite-dimensional division algebra and a subnormal subgroup of we show that if is a maximal subgroup of that satisfies a group identity,... 

In this thesis we study the structure of locally solvable, solvable, locally nilpotent, and nilpotent maximal subgroups of skew linear groups. In [5] it has been conjectured that if D is a division ring and M a nilpotent maximal subgroup of , then D is commutative. In connection with this conjecture we show that if M a nilpotent maximal subgroup of , then M is an abelian group. Also we show that is a solvable maximal subgroup of real quaternions and so give a counterexample to Conjecture 3 of [5], which states that if D is a division ring and M a solvable maximal subgroup of , then D is commutative. Also we completely determine the structure of division rings with a non-abelian... 

If L is a finite-dimensional Lie algebra over the field F then the universal enveloping algebra U(L) can be embedded in a division ring D. In particular, if L is a solvable p-algebra, there is a decomposition D=KR where K and R are maximal subfields of D, K is Galois extension of the center Z of D and R is a purely inseparable extension of Z with R^p⊆Z. The present thesis is concerned with the compared structures of maximal subfields in a division D and in the division ring of rational functions D(X). We prove that maximal subfields of D(X) “generically” specialize to maximal subfields of D, and properties such as being Galois or purely inseparable over the centre also carry over... 

In this Mater thesis, we o?er a general Prime Ideal Principle for proving that certain ideals in a commutative ring are prime. This leads to a direct and uniformtreatment of a number of standard results on prime ideals in commutative algebra,due to Krull, Cohen, Kaplansky, Herstein, Isaacs, McAdam, D.D. Anderson, andothers. More signi?cantly, the simple nature of this Prime Ideal Principle enablesus to generate a large number of hitherto unknown results of the “maximal impliesprime” variety. The key notions used in our uniform approach to such prime idealproblems are those of Oka families and Ako families of ideals in a commutative ring.In chapter 2, we amplify this study by developing... 

Defining the structure of maps using local features is among the popular fields of study in mathematics. Therefore determining the structure of maps on rings which behave like derivations at idempotent-product elements has been getting attention recently. This subject is useful for examining the structure of rings and algebraic operators in both algebra and analysis as well. Suppose that R is a ring, d : R ! R is an additive map, z 2 R and d meets the condition below: 8a; b 2 R : d(ab) = ad(b) + d(a)b Therefore d is called a derivation on R. If for every a; b 2 R where ab = z, d(ab) = ad(b) + d(a)b then d behaves like a derivation at idempotent-product elements of ab = z. The main challenge... 

In the present thesis, we study on the structure of the solvable, supersoluble, nilpotent, and irreducible structures of the subgroups. The main purpose of the present thesis is to represent a criterion given for D to be a supersoluble (nilpotent) crossed product division algebra in terms of subgroup of the multiplicative group D* of D. It is shown that the D is supersoluble (nilpotent) crossed product, if and only if D* contains an irreducible abelian-by-supersoluble (nilpotent) subgroup. Furtetmore, we review and discuss the structure of the crossed product division algebra, D, with the solvable irreducible subgroup, D*, and finally we extend our results for the semi-cross product of the... 

Given a finite dimensional F-central simple algebra A = Mn(D),the connection between the Ferattini subgroup (A) and (F) via Z(A′), the center of the derived group of A , is investigated. Setting G = F \(A), it is shown that (F)Z(A′ ) G (\pFp)Z(A′)where the intersection is taken over primes p not dividing the degree of A. Furthermore ,when F is a local or global field, the group G is completely determined. Using the above connection, (A) is also calculated for some particular division rings D  

This thesis is a survey of results of studies and researches in theory of Division rings and multiplicative groups of division rings and relations between group strcture and algebraic structure of division rings.in this case,we emphesized on study of finitely generated,maximal,nilpotent and soluble subgroups of division rings.we also,study the Valuation theory on division rings and Reduced K-theory of division rings and relations between these theories and group structure of division rings,morever,those which is finite dimensional over their center as vector spaces.at the end,we shortly,study divisible division rings and radicable division rings  

A Weddernurn polynomial over a division ring K, is the minimal polynomial of an algebraic subset of K. Such a polynomial, always is a product of linear factors over K, but not all such products are Wedderburn polynomials, even if these linear factors are distinct. In this thesis, we give some properties and characterizatios of Wedderburn polynomials over the division ring K, which relates deeply to algebraic subsets of K. We work in the general setting of Ore skew polynomials with an indeterminate t over K, corresponding to S,D, where S is an endomorphism of K and D is an S-derivation over K. Also we give a survey of the structure of the skew polynomial ring K[t; S; D] and its relation with... 

Our main purpose is introducing essential dimension and investigating properties of this concept and definition of it on different algebraic objects and proving some theories about it. In the beginning we define the concept of essential dimension on the extension fields which indeed it is expressing complexity of extension on the background field.Then with the meaning of noetherian extension which we will introduce it in the chapter three, we generalize the concept of essential dimension to finite groups. At last we investigate the connection between the essential dimension with generic polynomials and one of our important results is finding upper bounds for essential dimension of finite... 

Albert proposed the cyclic algebraof the degree 4 in1934. After that more studies were conducted on thecyclic algebras on F in conditions in which L = F(µ )nis an extention of F, for in the Albert’s example it wasmanifest that 2|[F(µ ) : F]. In a paper in the same yearnAlbert proposed a condition for an F-division algebrato be cyclic. In this thesis, a theorem will be proposedin which for the condition (n=1) the Albert’s theoremwill be the result. Moreover, F-division rings of the pndegree are investigated. the required means is modularspectral factorization which was for the ?rst time de?nedin clusters by Merkuryev. Finally a condition will beproposed for the fact that F-division... 

Let D be an F-central division algebra of index n. Here we present a criterion for the triviality of the group G(D) = D*=NrdD=F (D*)D′ and thus generalizing various related results published recently. To be more precise, it is shown that G(D) = 1 if and only if SK1(D) = 1 and F*2 = F*2n . Using this, we investigate the role of some particular subgroups of D* in the algebraic structure of D. In this direction, it is proved that a division algebra D of prime index is a symbol algebra if and only if D* contains a non-abelian nilpotent subgroup. More applications of this criterion including the computation of G(D) and the structure of maximal subgroups of D* are also investigated  

We survey the research conducted on zero divisor graphs, with a focus on zero divisor graphs determined by equivalence classes of zero divisors of a commutative ring R. In particular, we consider the problem of classifying star graphs with any finite number of vertices. We study the pathology of a zero divisor graph in terms of cliques, we investigate when the clique and chromatic numbers are equal, and we show that the girth of a Noetherian ring, if finite, is 3. We also introduce a graph for modules that is useful for studying zero divisor graphs of trivial extensions  

Given a divisible finite field extension K/F, the structure of Br(F), the Brauer group of F, is investigated. It is shown that, if F is indivisible, then Br(F) ≃ Z_2, which generalizes the Frobenius Theorem. As a consequence, when F is indivisible, the class of all finite dimensional non-commutative F-central division algebras D having radicable multiplicative groups is determined. In fact, it is proved that the following statements are equivalent: (1) D is radicable, (2) D contains a divisible subfield K/F, and (3) D is the ordinary quaternion division algebra and F(√-1) is divisible  

Let D be an F-central division algebra of index n. Here, we present a criterion for the triviality of the group G(D) = D∗/NrdD/F (D∗)D′. In fact, we show that G(D) = 1 if and only if F∗2 = F∗2n and SK1(D) = 1 where SK1(D) is the reduced whitehead group of D. Then, we use this criterion to investigate the role of (non-abelian) divisible groups in the structure of division algebras. We prove that if D is not the ordinary quaternion algebra, D is similar to a tensor product of F-central cyclic algebras and D∗ contains a (non-abelian) divisible maximal subgroup, then D = F. In this direction, we observe that if the index of D is a prime, then D is a symbol algebra if and only if D∗ contains a...