On Artinian Modules Over Duo Rings

Abstract

The well-known method, which reduces the study of Artinian modules over commutative rings to the study of Artinian modules over quasi-local rings, is extended to the study of these modules over duo rings. The duals of Krull Intersection Theorem and Nakayama's Lemma are proved for Artinian modules over a large class of duo rings. We also give an upper bound for the length of a duo ring R in terms of the length of a faithful R-module. This generalizes a well-known result of Schur concerning the cardinality of a maximal linearly independent set of commuting matrices.

Key Words:

• Artinian modules
• Attached prime ideals
• Duo ring
• Finitely embedded modules
• Krull intersection theorem
• Nakayama's lemma
• Schur's theorem

2000 Mathematics Subject Classification:

• 16P20
• 16A53
• 15A03
• 15A04

Notes

Communicated by T. Albu.