A. A. Tuganbaev

Maximal submodules and locally perfect rings

Rings over which every nonzero right module has a maximal submodule are calledright Bass rings. For a ringA module-finite over its centerC, the equivalence of the following conditions is proved:(1)A is a tight Bass ring;(2)A is a left Bass ring;(3)A/J(A) is a regular ring, andJ(A) is a right and leftt-nilpotent ideal.

Rings over Which each Module Possesses a Maximal Submodule

Right Bass rings are investigated, that is, rings over which any nonzero right module has a maximal submodule. In particular, it is proved that if any prime quotient ring of a ringA is algebraic over its center, thenA is a right perfect ring iffA is a right Bass ring that contains no infinite set of orthogonal idempotents.

Modules Over Serial Rings

This paper continues the study of Noetherian serial rings. General theorems describing the structure of such rings are proved. In particular, some results concerning π-projective and π-injective modules over serial rings are obtained.

Semidistributive and distributively decomposable rings

A module is said to be distributive if the lattice of all its submodules is distributive. A module is called semidistributive if it is a direct sum of distributive modules. Right semidistributive rings, as well as distributively decomposable rings, are investigated.

Flat modules and rings finitely generated as modules over their center

A module is called distributive (is said to be a chain module) if the lattice of all its submodules is distributive (is a chain). Let a ringA be a finitely generated module over its unitary central subringR. We prove the equivalence of the following conditions:(1)A is a right or left distributive semiprime ring;(2)for any maximal idealM of a subringR central inA, the ring of quotientsAM is a finite direct product of semihereditary Bézout domains whose quotient rings by the Jacobson radicals are finite direct products of skew fields;(3)all right ideals and all left ideals of the ringA are flat (right and left) modules over the ringA, andA is a distributive ring, without nonzero nilpotent elements, all of whose quotient rings by prime ideals are semihereditary orders in skew fields.

Modules over Endomorphism Rings

It is proved that A is a right distributive ring if and only if all quasiinjective right A-modules are Bezout left modules over their endomorphism rings if and only if for any quasiinjective right A-module M which is a Bezout left End (M)-module, every direct summand N of M is a Bezout left End(N)-module. If A is a right or left perfect ring, then all right A-modules are Bezout left modules over their endomorphism rings if and only if all right A-modules are distributive left modules over their endomorphism rings if and only if A is a distributive ring.

Bezout Rings, Polynomials, and Distributivity

AbstractLet A be a ring, ϕ be an injective endomorphism of A, and let

The Structure of Modules over Hereditary Rings

A_r \left[ {x,\varphi } \right] \equiv R