Two-sided residuation in the set of topologizing filters on a ring

摘要

The set Fil R-R of all right topologizing filters on a;fixed, but arbitrary ring R both a complete lattice under inclusion and a monoid with respect to an order compatible, but in general noncommutative, binary operation. It is known that he order dual. Fil R-R(du) of Fil R-R is always eft residuated, meaning, for each pair E, B is an element of Fil R-R, there exists a smallest H is an element of Fil R-R such that H : B superset of F but is not, in general, right residuated (there exists a smallest H such that B : H superset of F). This paper investigates rings R for which Fil R-R(du) is both left and right residuated. It is shown that every rightfully bounded Noetherian ring possesses this property and that a commutative ring will too if and only if the binary operation: oni Fil R-R is commutative. This has the c nsequence that if R valuation domain, then Fil R-R(du) will be left and right residtaated precisely hen f? is rank one discrete