Groups and Algebraicity in Complete Rank Rings

Introduction: It is well known that many of the results in classical linear algebra have an unequivocal extension to the more general situation when the scalars are drawn from an arbitrary division ring K. There are, however, three distinct theories of determinants for matrices over a division ring. One of these, originated by Study, "apples only to very particular non-communicative fields and to matrices of special type" (Dieudonne) and will not concern us here. The remaining two, one due to Ore and the other due to Dieudonne, reflect together, if not separately, the basic properties of the classical determinant. The diversity is, as we will see, due to the fact that the ordinary determinants for square matrices play different roles; fist, in connection with ideals in the matrix ring and then, if a matrix be non-singular, in connection with the group of invertible matrices.