COUNIFORM DIMENSION OVER SKEW POLYNOMIAL RINGS

ABSTRACT

In this paper, we study the behavior of the couniform (or dual Goldie) dimension of a module under various polynomial extensions. For a ring automorphism σ ∈ Aut(R), we use the notion of a σ-compatible module M _R to obtain results on the couniform dimension of the polynomial modules M[x], M[x ⁻¹], and M[x, x ⁻¹] over suitable skew extension rings.

Key Words:

• Annihilator-compliant polynomial
• Corank
• Couniform dimension
• Hollow module
• Inverse polynomial module
• Right perfect ring
• Sigma-compatible module
• Skew polynomial ring
• Uniform dimension

Mathematics Subject Classification:

• 16S36
• 16D10

ACKNOWLEDGMENT

I gratefully thank T. Y. Lam, Greg Marks, and the referee for their many helpful suggestions, comments, and ideas related to this article. I also thank my parents, Arthur and Juliann Annin, for the hospitality they offered me at their home while I was working on this paper.

Notes

¹Other synonyms for the uniform dimension include Goldie dimension and Goldie rank.

²Recall that a submodule S _R of an arbitrary module M _R is called small if, for every submodule N _R  ⊆ M _R with N + S = M, we have N = M. In this case, we write S _R  ⊆  _s M _R .

³The radical of a module M _R , written rad(M _R ), is defined to be the intersection of all maximal submodules of M _R . Pertinent facts on the radical can all be found in Chapter 24 of Lam (1991).

⁴Recall that the existence of such a polynomial hinges on the compatibility assumption on M _R .

⁵Recall that a ring R is a right perfect ring if and only if R/radhskip3ptR is semisimple and every nonzero module M _R contains a maximal submodule (Lam (1991), Exercise 24.3).

Communicated by M. Ferroro.