On natural sets on an idiom

Abstract

We introduce and develop the concept of natural set for an idiom (complete upper-continuous modular lattice), which is the analogous of the well-known natural classes of modules. For these natural sets, we generalize important results of the natural classes of modules, and get relevant information about the idiom. In particular, we characterize when the frame of nucleus on an idiom is Boolean. We also apply these results to classes of modules, proving when R–tors and R–nat are isomorphic, only studying the structure of $\Lambda (R),$ as an idiom, and its frame of nucleus.

Keywords:

• Idiom
• frame
• lattice
• natural set
• natural class
• R–tors
• semiartinian ring
• semiartinian lattice

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 06B80
• 16D80
• 16S99
• 06B75
• 06D99
• 06C22

Acknowledgment

The author wish to express his gratitude to the referee for his valuable remarks and suggestions.

Additional information

Funding

The author was supported by The National Council of Science and Technology (CONACYT) of Mexico with a scholarship for a Postdoctoral Stay in the University of Granada.