Group rings and rings of matrices.(English)Zbl 1136.20004

The author proves that the group ring $$RG$$ of a group $$G$$ of order $$n$$ over a ring $$R$$ is isomorphic to a certain ring of $$n\times n$$ matrices over $$R$$. Let $$w=\sum_{i=1}^n\alpha_{g_i}g_i\in RG$$, where $$g_i\in G$$, $$\alpha_{g_i}\in R$$. Then $$g_jw=\sum_{i=1}^n\alpha_{g_i}g_jg_i=\sum_{k=1}^n\alpha_{g_j^{-1}g_k}g_k$$. Thus the element $$w\in RG$$ corresponds to the $$n\times n$$ matrix $$M(RG,w)$$ which has the quantity $$\alpha_{{g_j^{-1}}g_k}\in R$$ as the element in the $$j$$-th row and the $$k$$-th column.
The main theorem in the paper establishes that the mapping $$\sigma\colon w\mapsto M(RG,w)$$ determines a bijective ring homomorphism between $$RG$$ and a subring of the matrix ring. A total of twelve theorems, dedicated to the study of the structure of $$\text{Im\,}\sigma$$, is proved. The cases when $$G$$ is either cyclic, elementary Abelian, dihedral or a direct product of cyclic groups are examined. The last two theorems deal with the case when $$G$$ is infinite. The author provides a criterion when an element of $$RG$$ is a zero-divisor, and when it is invertible. If $$G$$ is a locally finite group and $$R$$ is a field, then the finitely generated torsion group of $$U(RG)$$ is finite, i.e. the generalised Burnside problem has a positive answer for $$U(RG)$$.

MSC:

 20C05 Group rings of finite groups and their modules (group-theoretic aspects) 20C07 Group rings of infinite groups and their modules (group-theoretic aspects) 16S34 Group rings 15A30 Algebraic systems of matrices 16S50 Endomorphism rings; matrix rings 16U60 Units, groups of units (associative rings and algebras)

Keywords:

group rings; rings of matrices