Narrow Results

In this paper, we prove that every matrix over a division ring is representable as a product of at most 10 traceless matrices as well as a product of at most four semi-traceless matrices. By applying this result and the obtained so far other results, we show that elements of some algebras possess some rather interesting and nontrivial decompositions into products of images of non-commutative polynomials.

We study structures on the fields of characteristic zero obtained by introducing (multi-valued) operations of raising to power. Using Hrushovski–Fraisse construction we single out among the structures exponentially-algebraically closed once and prove, under certain Diophantine conjecture, that the first order theory of such structures is model complete and every its completion is superstable.

Let F be a field and G an Abelian group. For every prime number q and every ordinal number α we compute only in terms of F and G the Warfield q-invariants W_{α, q}(VF[G]) of the group VF[G] of all normed units in the group algebra F[G] under some minimal restrictions on F and G.

This expands own recent results from (Extracta Mathematicae, 2005) and (Collectanea Mathematicae, 2008).

We determine entirely which Artinian rings have maximal subring. In particular, we show that an Artinian ring without maximal subring is integral over some finite subring and in particular that every Artinian ring which is uncountable or of characteristic zero has a maximal subring. We also determine when a finite direct product of rings has a maximal subring. Finally, we show that if a ring R has an Artinian maximal subring then R itself is Artinian.