Direct sums of \(R\)-vector spaces (Q1908941)

The results of \textit{F. O. Stroup} [On the theory of Boolean vector spaces. Doctoral Thesis, Univ. of Missouri, Columbia (1969)] on Boolean vector spaces are extended by the author to vector spaces over a regular ring \(R\) through studying the concept of a direct sum of vector spaces over this regular ring \(R\). As the main result of his research, he proves that: Let \(G^*\) be the set of all nonzero-elements of the group \(G\), then \((\sum^n_{i = 1} G_i)^*\) is a basis for \(\sum^n_{i = 1} V_i\) if \(V_1, \dots, V_n\) are vector spaces over the same regular ring \(R\) having bases \(G_1^*, \dots, G^*_n\), respectively, where \(\sum^n_{i = 1} G_i\) is considered as the direct sum of additive groups. Moreover, if \(V_1, \dots, V_n\) are abstract \(R\)-vector spaces of finite dimensions \(d_1, \dots, d_n\), respectively, then \(\sum^n_{i = 1} V_i\) has dimension equal to \(\prod^n_{i = 1} (d_i + 1) - 1\).