Inverses and Zero Divisors in Matrix Rings

1. Introduction THE following question was mentioned by van der Waerden (1): ' If R is a ring with a unit element and A is an n X n square matrix with elements in R, can A, regarded as an element of the ring of nxn square matrices with elements in R, be both a left inverse and a left divisor of zero ? In other words can there exist nxn square matrices, A, B, X such that, denoting by / and 0 respectively the nxn unit and zero matrices, the conditions AB = I, AX = O, X ^ O are satisfied? This can clearly happen, for the value n = 1, if the ring R possesses an element which is both a left inverse and a left divisor of zero in R. We must therefore add the additional condition that the ring R does not possess elements a, 6, a; such that ab = e, ax = 0, x ^ 0 (e being the unit element of R). If the ring R is subjected to various additional conditions, then the non-existence of such matrices A can be proved. For example, if R is commutative such a matrix cannot exist since its determinant would have to be a left inverse and divisor of zero in R. Nor can they exist if R is finite or, more generally, if either the ascending or descending chain condition for left ideals is satisfied in R (see Baer, 2). The same is true if R can be immersed in a skew-field (see, for example, Lesieur (3), who proves a somewhat more general result). In this paper it will be shown that if no additional conditions are imposed on R, then such matrices can exist. We shall prove