Recognition and computations of matrix rings
This paper continues the study of criteria for a ring \(R\) to be a full \(n\) by \(n\) matrix ring \(M_n(A)\) over some ring \(A\) and methods for constructing such a base-ring \(A\) (which is not in general unique up to isomorphism). For instance it was shown by Agnarsson, Amitsur and Robson that if \(m\) and \(n\) are positive integers then \(R\) is isomorphic to \(M_{m+n}(A)\) for some ring \(A\) if and only if \(R\) has elements \(a,b,f\) such that \(f^{m+n}=0\) and \(af^m+f^nb=1\); it is shown here that, when this happens, \(A\) can be taken to be the eigenring \(E_R(fR)\) which, by definition, is \(I(fR)/fR\) where \(I(fR)\) is the idealiser of the right ideal \(fR\) of \(R\). These results are used to show that certain (factor rings) of Ore extensions are matrix rings and to construct corresponding base-rings as eigenrings. Also, many new criteria are given for \(R\) to be a full \(n\) by \(n\) matrix ring, several of them involving the existence of an \(n\)-th root for a matrix over \(R\). For example, one criterion is that there exists a positive integer \(r\neq q\) such that the \(r\) by \(r\) matrix with 1's immediately below the main diagonal and 0's elsewhere should have an \(n\)-th root in \(M_r(R)\). Finally, the authors extend a result of Levy, Robson and Stafford on ``hidden matrices'' by showing that if \(H\) is the ring of integer quaternions and \(c\) is a positive integer, then \(H+M_2(cH)\) is a full 2 by 2 matrix ring if and only if all the prime factors of \(c\) are congruent to 1 modulo 4.
scientific article; zbMATH DE number 997595
scientific article