Krull Dimension and Classical Krull Dimension of Modules

Abstract

Using the concept of prime submodule defined by Raggi et al. in [16], for M ∈ R-Mod we define the concept of classical Krull dimension relative to a hereditary torsion theory τ ∈M-tors. We prove that if M is progenerator in σ[M], τ ∈M-tors such that M has τ-Krull dimension then cl.K _τdim (M) ≤ k _τ(M). Also we show that if M is noetherian, τ-fully bounded, progenerator of σ[M], and M ∈ 𝔽_τ, then cl·K _τdim (M) = k _τ(M).

Key Words:

• Classical Krull dimension
• Krull dimension
• Prime submodules
• Semiprime submodules
• Torsion theory

2010 Mathematics Subject Classification:

• 16S90
• 16D50
• 16P50
• 16P70

Acknowledgments

Dedicated to the memory of Professor Francisco Raggi.

Notes

Communicated by T. Albu.