Some new dimensions of modules and rings

ABSTRACT

We say that a class 𝒫 of right modules over a fixed ring R is an epic class if it is closed under homomorphic images. For an arbitrary epic class 𝒫, we define a 𝒫-dimension of modules that measures how far modules are from the modules in the class 𝒫. For an epic class 𝒫 consisting of indecomposable modules, first we characterize rings whose modules have 𝒫-dimension. In fact, we show that every right R-module has 𝒫-dimension if and only if R is a semisimple Artinan ring. Then we study fully Hopfian modules with 𝒫-dimension. In particular, we show that a commutative ring R with 𝒫-dimension (resp. finite 𝒫-dimension) is either local or Noetherian (resp. Artinian). Finally, we show that Mat_m(R) is a right Köthe ring for some m if and only if every (left) right module is a direct sum of modules of 𝒫-dimension at most n for some n, if and only if R is a pure semisimple ring.

KEYWORDS:

• Local dimension
• pure semisimple ring
• 𝒫-dimension
• semisimple Artinian ring
• uniserial dimension

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• Primary: 16D70
• 16D10
• 16P70
• 16P20
• Secondary: 03E10

Acknowledgments

The authors would like to thank the associate editor, Professor Alberto Facchini, and the anonymous referee for their careful reading and comments, which helped to significantly improve the paper.