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Second Modules Over Noncommutative Rings
- Source :
- Communications in Algebra. 41:83-98
- Publication Year :
- 2013
- Publisher :
- Informa UK Limited, 2013.
-
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...
Details
- ISSN :
- 15324125 and 00927872
- Volume :
- 41
- Database :
- OpenAIRE
- Journal :
- Communications in Algebra
- Accession number :
- edsair.doi...........59c31030ab12cac34eb6048a436e0e41