Topologies on the quotient field of a Dedekind domain

Abstract

It is well known that if D is a Dedekind domain with quotient fieldK and if T is any Hausdorff nondiscrete field topology on K for which the open D-submodules of K form a fundamental system of neighborhoods of zero, then T is the supremum of a family of /?-adic topologies. We show that if the class number of K over D is finite and if T is any Hausdorff nondiscrete field topology on K for which D is a bounded set, then T is the supremum of a family of /?-adic topologies. We then investigate the problem of extending a locally bounded topology from D to a locally bounded topology on K. The extendable topologies on D for which there exists a nonzero topological nilpotent and for which D is a bounded set are characterized. Moreover it is shown that the topology of a locally compact principal ideal domain A extends to a ring topology on the quotient field of A if and only if A is compact.

The classification of compact right chain rings

“…to you by | University of Glasgow Library Authenticated Download Date | 6/25/15 4:56 AM In the commutative case, a compact chain domain is equivalent to a compact principal ideal domain by 4.1, and hence is a Dedekind domain by [51; page 270]. Thus by[10] or[41], a compact commutative chain domain is openly embeddable in We next show that this result also holds in the non-commutative case. §5.…”

“…Now a right chain domain R is a right Ore domain and so possesses a right skewfield of fractions Q(R). In the commutative case Jo-Ann Cohen [10] and independently Schiffels-Stenzel [41] showed that a compact principal ideal domain (or equivalently an invariant chain domain) is always openly embeddable in Q (R). In section five we show that an infinite compact ring R is a right chain domain if and only if R is a right Ore domain which is openly embeddable in Q(R) (necessarily a nondiscrete locally compact totally disconnected skewfield) such that the radical of Fn /?, where F is the centre of Q (/?…”