Distributive non-localizable rings

Abstract

All rings are associative with a non-zero identity. A module is called distributive if the lattice of all its submodules is distributive. A module in which all finitely generated submodules are cyclic is called a Bezout module. A module is called a chain module if any two of its submodules are comparable with respect to inclusion. A module is called uniform if any two of its non-zero submodules have a non-zero intersection. A ring is called right invariant if all its right ideals are ideals. A ring A is called right coherent if each of its finitely generated right ideals is a finitely presentableA-module. Expressions such as “distributive ring” mean that the corresponding conditions are satisfied on the right and left. For a ring A we denote by J(A), N (A), Nr(A), max(AA), and max(AA) the Jacobson radical, the set of all left zero divisors, the set of all right zero divisors, the set of all maximal right ideals, and the set of all maximal left ideals, respectively. A ring A is called right localizable if for any M ∈ max(AA) there is a right localization AM (that is, there are a ring AM and a ring homomorphism: f : A → AM such that all elements of f(A\M) are invertible in AM , AM = {f(a)f(t)−1 | a ∈ A, t ∈ A \M}, and Ker(f) = {a ∈ A | (A \M) ∩ rA(a) = ∅}).