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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)
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