Pavel Příhoda

Endomorphism Rings with Finitely Many Maximal Right Ideals

We show that the indecomposable R-modules whose endomorphism ring has finitely many maximal right ideals, all of them two-sided, have a surprisingly simple behavior as far as direct sums are concerned. Our main result is that these modules are completely described up to isomorphism by an easy combinatorial structure, a simple hypergraph. If 𝒞 is any full subcategory of Mod-R containing all these modules as objects, the vertices of the hypergraph are suitable ideals 𝒫 of the category 𝒞. Let SFT-R be the category of all finite direct sums of modules whose endomorphism ring has finitely many maximal right ideals. The objects of SFT-R are completely determined up to isomorphism by the dimensions of vector spaces indexed by suitable ideals 𝒫 of the category SFT-R. Several examples are given in the last section.

A Version of the Weak Krull–Schmidt Theorem for Infinite Direct Sums of Uniserial Modules

We show a version of the weak Krull–Schmidt theorem concerning infinite families of uniserial modules.

Big projective modules over noetherian semilocal rings

Abstract We prove that for a noetherian semilocal ring R with exactly k isomorphism classes of simple right modules the monoid V*(R) of isomorphism classes of countably generated projective right (left) modules, viewed as a submonoid of V*(R/J(R)), is isomorphic to the monoid of solutions in (ℕ0 ∪ {∞}) k of a system consisting of congruences and diophantine linear equations. The converse also holds, that is, if M is a submonoid of (ℕ0 ∪ {∞}) k containing an order unit (n 1, . . . , nk ) of which is the set of solutions of a system of congruences and linear diophantine equations then it can be realized as V*(R) for a noetherian semilocal ring such that R/J(R) ≅ Mn1 (D 1) × ⋯ × Mnk (Dk ) for suitable division rings D 1, . . . , Dk .

Infinitely generated projective modules over pullbacks of rings

We use pullbacks of rings to realize the submonoids

ITERATED POWER INTERSECTIONS OF IDEALS IN RINGS OF ITERATED DIFFERENTIAL POLYNOMIALS

M

Projective modules are determined by their radical factors

of

Weak Krull–Schmidt theorem and direct sum decompositions of serial modules of finite Goldie dimension

(\N_0\cup\{\infty\})^k

The Krull-Schmidt Theorem in the Case Two

which are the set of solutions of a finite system of linear diophantine inequalities as the monoid of isomorphism classes of countably generated projective right

On uniserial modules that are not quasi-small

R

Classifying generalized lattices: some examples as an introduction

-modules over a suitable semilocal ring. For these rings, the behavior of countably generated projective left