Dimension on non-essential submodules

In this paper, we introduce and study the concepts of non-essential Krull dimension and non-essential Noetherian dimension of an [Formula: see text]-module, where [Formula: see text] is an arbitrary associative ring. These dimensions are ordinal numbers and extend the notion of Krull dimension. They respectively rely on the behavior of descending and ascending chains of non-essential submodules. It is proved that each module with non-essential Krull dimension (respectively, non-essential Noetherian dimension) has finite Goldie dimension. We also show that a semiprime ring [Formula: see text] with non-essential Noetherian dimension is uniform.

Avoiding modules, co-avoiding modules, Goldie dimension and its dual

We introduce and study two concepts in module theory which are dual to each other. It will be revealed that, in the finite case, they are new equivalent concepts for the Goldie dimension of modules and the dual Goldie dimension. However, we shall observe, in the infinite case, they are different concepts.

GOLDIE DIMENSION OF CONSTANTS OF LOCALLY NILPOTENT SKEW DERIVATIONS

In this paper, we examine rings R with locally nilpotent skew derivations d and compare the Goldie dimension of R to that of the subring of constants Rd. This generalizes the situation where one compares the Goldie dimension of an Ore extension to that of the base ring. Under certain natural conditions placed upon Rd, we show that R and Rd have the same Goldie dimension.

RINGS WHOSE CYCLIC MODULES ARE DIRECT SUMS OF EXTENDING MODULES

Dedekind domains, Artinian serial rings and right uniserial rings share the following property: Every cyclic right module is a direct sum of uniform modules. We first prove the following improvement of the well-known Osofsky-Smith theorem: A cyclic module with every cyclic subfactor a direct sum of extending modules has finite Goldie dimension. So, rings with the above-mentioned property are precisely rings of the title. Furthermore, a ring R is right q.f.d. (cyclics with finite Goldie dimension) if proper cyclic (≇ RR) right R-modules are direct sums of extending modules. R is right serial with all prime ideals maximal and ∩n ∈ ℕJn = Jm for some m ∈ ℕ if cyclic right R-modules are direct sums of quasi-injective modules. A right non-singular ring with the latter property is right Artinian. Thus, hereditary Artinian serial rings are precisely one-sided non-singular rings whose right and left cyclic modules are direct sums of quasi-injectives.

ON PRIME AND SEMIPRIME GOLDIE MODULES

A right R-module M is called a Goldie module if it has finite Goldie dimension and satisfies the ACC for M-annihilator submodules of M. In this paper, we study the class of prime Goldie modules and the class of semiprime Goldie modules as generalizations of prime right Goldie rings and semiprime right Goldie rings.