Second Modules Over Noncommutative Rings

Abstract

Let R be an arbitrary ring. A nonzero unital right R-module M is called a second module if M and all its nonzero homomorphic images have the same annihilator in R. It is proved that if R is a ring such that R/P is a left bounded left Goldie ring for every prime ideal P of R, then a right R-module M is a second module if and only if Q = ann _R (M) is a prime ideal of R and M is a divisible right (R/Q)-module. If a ring R satisfies the ascending chain condition on two-sided ideals, then every nonzero R-module has a nonzero homomorphic image which is a second module. Every nonzero Artinian module contains second submodules and there are only a finite number of maximal members in the collection of second submodules. If R is a ring and M is a nonzero right R-module such that M contains a proper submodule N with M/N a second module and M has finite hollow dimension n, for some positive integer n, then there exist a positive integer k ≤ n and prime ideals P _i  (1 ≤ i ≤ k) such that if L is a proper submodule of M with M/L a second module, then M/L has annihilator P _i for some 1 ≤ i ≤ k. Every second submodule of an Artinian module is a finite sum of hollow second submodules.

Key Words:

• Attached prime ideal
• Hollow dimension
• Second module
• Semilocal rings

2000 Mathematics Subject Classification:

• 16D10
• 16N60
• 16L30
• 16L60
• 16L99

ACKNOWLEDGMENT

The first and second authors were supported by the Scientific Research Project Administration of Akdeniz University. We thank the referee for careful reading of the manuscript.

Notes

Communicated by T. Albu.