Normal subgroups in the group of column-finite infinite matrices

The classical result, due to Jordan, Burnside, Dickson, says that every normal subgroup of GL ⁢ ( n , K ) \mathrm{GL}(n,K) , where 𝐾 is a field and n ≥ 3 n\geq 3 , which is not contained in the center contains SL ⁢ ( n , K ) \mathrm{SL}(n,K) . Rosenberg described the normal subgroups of GL ⁢ ( V ) \mathrm{GL}(V) , where 𝑉 is a vector space of any infinite cardinality dimension over a division ring. However, when he considers subgroups of the direct product of the center and the group of linear transformations 𝑔 such that g - id V g-\mathrm{id}_{V} has finite-dimensional range, the proof is incomplete. We fill this gap for countably dimensional 𝑉 giving description of the lattice of normal subgroups in the group of infinite column-finite matrices indexed by positive integers over any field.

The universality of Hughes-free division rings

Let $$E*G$$ E ∗ G be a crossed product of a division ring E and a locally indicable group G. Hughes showed that up to $$E*G$$ E ∗ G -isomorphism, there exists at most one Hughes-free division $$E*G$$ E ∗ G -ring. However, the existence of a Hughes-free division $$E*G$$ E ∗ G -ring $${\mathcal {D}}_{E*G}$$ D E ∗ G for an arbitrary locally indicable group G is still an open question. Nevertheless, $${\mathcal {D}}_{E*G}$$ D E ∗ G exists, for example, if G is amenable or G is bi-orderable. In this paper we study, whether $${\mathcal {D}}_{E*G}$$ D E ∗ G is the universal division ring of fractions in some of these cases. In particular, we show that if G is a residually-(locally indicable and amenable) group, then there exists $${\mathcal {D}}_{E[G]}$$ D E [ G ] and it is universal. In Appendix we give a description of $${\mathcal {D}}_{E[G]}$$ D E [ G ] when G is a RFRS group.

Exploring the Beginnings of Algebraic K-Theory

According to Atiyah, K-theory is that part of linear algebra that studies additive or abelian properties (e.g. the determinant). Because linear algebra, and its extensions to linear analysis, is ubiquitous in mathematics, K-theory has turned out to be useful and relevant in most branches of mathematics. Let R be a ring. One defines K0(R) as the free abelian group whose basis are the finitely generated projective R-modules with the added relation P ⊕ Q = P + Q. The purpose of this thesis is to study simple settings of the K-theory for rings and to provide a sequence of examples of rings where the associated K-groups K0(R) get progressively more complicated. We start with R being a field or a principle ideal domain and end with R being a polynomial ring on two variables over a non-commutative division ring.

On γ-Sets in Rings

Let R be a ring with identity 1R. A subset J of R is called a γ-set if for every a ∈ R\J,there exist b, c ∈ J such that a+b = 0 and ac = 1R = ca. A γ-set of minimum cardinality is called a minimum γ-set. In this study, we identified some elements of R that are necessarily in a γ-sets, and we presented a method of constructing a new γ-set. Moreover, we gave: necessary and sufficient conditions for rings to have a unique γ-set; an upper bound for the total number of minimum γ-sets in a division ring; a lower bound for the total number of minimum γ-sets in a division ring; necessary and sufficient conditions for T(x) and T to be equal; necessary and sufficient conditions for a ring to have a trivial γ-set; necessary and sufficient conditions for an image of a γ-set to be a γ-set also; necessary and sufficient conditions for a ring to have a trivial γ-set; and, necessary and sufficient conditions for the families of γ-sets of two division rings to be isomorphic.

Properties of the commutators of some elements of linear groups over divisions rings

Inclusions resulting from the commutativity of elements and their commutators with trans\-vections in the language of residual and fixed submodules are found. The residual and fixed submodules of an element $\sigma $ of the complete linear group are defined as the image and the kernel of the element $\sigma -1$ and are denoted by $R(\sigma )$ and $P(\sigma )$, respectively. It is shown that for an arbitrary element $g$ of a complete linear group over a division ring whose characteristic is different from 2 and the transvection $\tau $ from the commutativity of the commutator $\left[g,\tau \right]$ with $g$ is followed by the inclusion of $R(\left[g,\tau \right])\subseteq P(\tau )\cap P(g)$. It is proved that the same inclusions occur over an arbitrary division ring if $g$ is a unipotent element, $\mathrm{dim}\mathrm{}(R\left(\tau \right)+R\left(g\right))\le 2$ and the commutator $\left[g,\tau \right]$ commutes with $\tau $ or if $g$ is a unipotent commutator of some element of the complete linear group and transvection $\ \tau $.