caa a22 4500 386385955 CHVBK 20180307112058.0 cr unu---uuuuu 161130e198901 xx s 000 0 eng 10.1017/S0305004100001365 doi S0305004100001365 pii (NATIONALLICENCE)cambridge-10.1017/S0305004100001365 Chatters A. W. University of Bristol, BS8 1TW Representation of tiled matrix rings as full matrix rings [Elektronische Daten] [A. W. Chatters] It can be very difficult to determine whether or not certain rings are really full matrix rings. For example, let p be an odd prime, let H be the ring of quaternions over the integers localized at p, and set Then T is not presented as a full matrix ring, but there is a subring W of H such that T M2(W). On the other hand, if we take H to be the ring of quaternions over the integers and form T as above, then it is not known whether T M2(W) for some ring W. The significance of p being an odd prime is that H/pH is a full 2 x 2 matrix ring, whereas H/2H is commutative. Whether or not a tiled matrix ring such as T above can be re-written as a full matrix ring depends on the sizes of the matrices involved in T and H/pH. To be precise, let H be a local integral domain with unique maximal ideal M and suppose that every one-sided ideal of H is principal. Then H/M Mk(D) for some positive integer k and division ring D. Given a positive integer n. let T be the tiled matrix ring consisting of all n x n matrices with elements of H on and below the diagonal and elements of M above the diagonal. We shall show in Theorem 2.5 that there is a ring W such that T Mn(W) if and only if n divides k. An important step in the proof is to show that certain idempotents in T/J(T) can be lifted to idempotents in T, where J(T) is the Jacobson radical of T. This technique for lifting idempotents also makes it possible to show that there are (k + n − 1)!/ k!(n−1)! isomorphism types of finitely generated indecomposable projective right T-modules (Theorem 2·10). Copyright © Cambridge Philosophical Society 1989 Mathematical Proceedings of the Cambridge Philosophical Society Cambridge University Press 105/1(1989-01), 67-72 0305-0041 105:1<67 1989 105 PSP https://doi.org/10.1017/S0305004100001365 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1017/S0305004100001365 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Chatters A. W. University of Bristol, BS8 1TW NATIONALLICENCE 773 0- Mathematical Proceedings of the Cambridge Philosophical Society Cambridge University Press 105/1(1989-01), 67-72 0305-0041 105:1<67 1989 105 PSP CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-cambridge