Question Of Kaplansky Research Articles

On a question of Kaplansky

Kaplansky [7] proved that CK(X) is the intersection of all free maximal ideals in C(X) in the case of discrete X, and asked whether the equality holds in general. In this paper we prove that CK(X) coincides with the intersection of all free maximal ideals if and only if every open hemicompact z-compact (i.e., every zero-set contained in it is compact) subset of X is relatively compact or equivalently, every open Lindelöf z-compact subset of X is relatively compact. We conclude that the equality holds whenever X is a strongly isocompact space.

Local duo rings whose finitely generated modules are direct sums of cyclics

In this paper, we give an answer to the following question of Kaplansky [14] in the local case: For which duo rings R is it true that every finitely generated left R-module can be decomposed as a direct sum of cyclic modules? More precisely, we prove that for a local duo ring R, the following are equivalent: (i) Every finitely generated left R-module is a direct sum of cyclic modules; (ii) Every 2-generated left R-module is a direct sum of cyclic modules; (iii) Every factor module of R R ⊕ R is a direct sum of cyclic modules; (iv) Every factor module of R R ⊕ R is serial; (v) Every finitely generated left R-module is serial; (vi) R is uniserial and for every non-zero ideal I of R, R/I is a linearly compact left R-module; (vii) R is uniserial and every indecomposable injective left R-module is left uniserial; and, (viii) Every finitely generated right R-module is a direct sum of cyclic modules.

On a Question of Kaplansky

On a Question of Kaplansky

On a Question of Kaplansky

On a Question of Kaplansky

Discontinuous Homomorphisms from Non-Commutative Banach Algebras

In the 1970s, a question of Kaplansky about discontinuous homomorphisms from certain commutative Banach algebras was resolved. Let A be the commutative C*-algebra C(Ω), where Ω is an infinite compact space. Then, if the continuum hypothesis (CH) be assumed, there is a discontinuous homomorphism from C(Ω) into a Banach algebra [2, 7]. In fact, let A be a commutative Banach algebra. Then (with (CH)) there is a discontinuous homomorphism from A into a Banach algebra whenever the character space ΦA of A is infinite [3, Theorem 3] and also whenever there is a non-maximal, prime ideal P in A such that | A / P | = 2 ℵ 0 [ 4 , 8 ] . (It is an open question whether or not every infinite-dimensional, commutative Banach algebra A satisfies this latter condition.) 1991 Mathematics Subject Classification 46H40.