Isomorphism of generalized matrix rings

An isomorphism problem is considered for generalized matrix rings with values in a given ring R. An exhaustive answer is given for the case of a commutative domain R and a commutative local ring R.     

Isomorphism of complete local noetherian rings and strong approximation

About a year ago Angus Macintyre raised the following question. Let and be complete local noetherian rings with maximal ideals and such that is isomorphic to for every . Does it follow that and are isomorphic? We show that the answer is yes if the residue field is algebraic over its prime field. The proof uses a strong approximation theorem of Pfister and Popescu, or rather a variant of it, which we obtain by a method due to Denef and Lipshitz. Examples by Gabber show that the answer is no in general.

     

FGF group rings

Periodica Mathematica Hungarica - A ring R is called a left FGF ring if every finitely generated left R-module can be embedded in a free left R-module. It is proved that a group ring RG is left FGF...     

Armendariz group rings

For a torsion or torsion-free group G and a field F, we characterize the group algebra FG that is Armendariz. Armendariz property for a group ring over a general ring R is also studied and related to those of Abelian group rings and the quaternion ring over R.     

Morphic group rings

An element a in a ring R is called left morphic if there exists b∈R such that l_R(a)=Rb and l_R(b)=Ra, where l_R(a) denotes the left annihilator of a in R. The ring R is called left morphic if every element of R is left morphic. Left morphic rings have been studied by Nicholson and Sánchez Campos. In this paper, the question of when a group ring is left morphic is discussed in great detail and various morphic group rings are identified.     

Duo group rings

It is shown that the group algebra of a torsion group G over a field K is duo if and only if it is reversible.     

Twisted group rings and quasi-frobenius quotient rings

Archiv der Mathematik -     

Endomorphism rings of simple modules over group rings

If is a finitely generated nilpotent group which is not abelian-by-finite, a field, and a finite dimensional separable division algebra over , then there exists a simple module for the group ring with endomorphism ring . An example is given to show that this cannot be extended to polycyclic groups.

     

On embedding of group rings in division rings

LetF be a free group,N⊲F andV(N) be a verbal subgroup ofN. For the group ringR , whereR is any field and F/V(N), the zero divisor problem of Kaplansky and the problem of embeddingR in a division ring are investigated. It is proved, in particular, thatR has no zero divisors and can be embedded in a division ring whenF/N is finitely approximated andN/V(N) is approximated by nilpotent groups without torsion.