Orders in locally pro-Artinian rings

Abstract

Compact open subrings of a simple algebra over a p-field, i.e., a locally compact, non-discrete, non-connected field, are called orders. In analogy to the concept of local compactness we shall call a topological ring A locally pro-Artinian, if it contains an open (left-) pro-Artinian subring. Such an open pro-Artinian subring we shall call an order of A, provided A has no open left ideals. It should be recalled from [2, (5. l)] that a topological ring A without open left ideals containing an open subring R is completely determined by R. In fact A = Q&R) = Q,(R), the ring of quotients with respect to the filter 9 of open left ideals of R. Furthermore 9 is a perfect topology in the sense of [13] and R is P-torsion-free. Q,(R) denotes the topological ring of quotients discussed in [2]. Theorem 1 of the paper gives a complete description of orders in a commutative, locally pro-Artinian ring without open ideals. Theorem 2 extends these results to a special class of left-pro-Artinian rings.