Local domains with topologically<i>T</i>-nilpotent radical

Abstract

A local ring A will be said to have TTN (respectively Krull dimension 1) if M(A) is topologically T-nilpotent (respectively topologically nilpotent.) It is clear that "TTN" is stronger than "Krull dim 1". EXAMPLES. If K is a field, if[[X]], the ring of formal power series in one indeterminant, is a local domain with TTN. Formal series in more indeterminants are local domains with neither TTN or Krull dim 1. More generally, a discrete valuation ring has TTN. EXAMPLE. Let if be a field, G the set of nonnegative real numbers under addition. Then K G is a valuation ring with Krull dim 1 but not TTN. 2. Relationships to valuation rings* The following theorem is well-known. THEOREM. Let A be a local domain, K its field of fraction. Then there is a valuation v on K such that A is dominated by the valuation ring of v. (See [4], p. 92). PROPOSITION. Let E and F be A-modules, A a local domain with TTN. If E&) F