On subdirect product of prime modules
2
조회
0
다운로드
0
인용
0
참고문헌
키워드
초록
In the various module generalizations of the concepts of prime (semiprime) for a ring, the question ``when are semiprime modules subdirect product of primes?" is a serious question in this context and it is considered by earlier authors in the literature. We continue study on the above question by showing that: If $R$ is Morita equivalent to a right pre-duo ring (e.g., if $R$ is commutative) then weakly compressible $R$-modules are precisely subdirect products of prime $R$-modules if and only if ${\rm dim}(R)=0$ and $R/{\rm N}(R)$ is a semi-Artinian ring if and only if every classical semiprime module is semiprime. In this case, the class of weakly compressible $R$-modules is an enveloping for Mod-$R$. Some related conditions are also investigated.
Abstract
In the various module generalizations of the concepts of prime (semiprime) for a ring, the question ``when are semiprime modules subdirect product of primes?" is a serious question in this context and it is considered by earlier authors in the literature. We continue study on the above question by showing that: If $R$ is Morita equivalent to a right pre-duo ring (e.g., if $R$ is commutative) then weakly compressible $R$-modules are precisely subdirect products of prime $R$-modules if and only if ${\rm dim}(R)=0$ and $R/{\rm N}(R)$ is a semi-Artinian ring if and only if every classical semiprime module is semiprime. In this case, the class of weakly compressible $R$-modules is an enveloping for Mod-$R$. Some related conditions are also investigated.
참고문헌 (17)
- F. W. Anderson, "Rings and categories of modules" (1973), Springer
- O. D. Avraamova, "Abelian groups and modules, No. 8", pp.3~16 (1989), Tomsk. Gos. Univ.
- M. Behboodi, "A generalization of Baer’s lower nilradical for modules", J. Algebra Appl, 6(2), pp.337~353 (2007)
- L. Bican, "Prime and coprime modules", Fund. Math, 57(1), pp.33~45 (1980)
- N. Dehghani, "A characterization of modules embedding in products of primes and enveloping condition for their class", J. Algebra Appl, 14(4), pp.14~ (2015)
- E. Enochs, "The existence of envelopes", Rend. Sem. Mat. Univ. Padova, 90, pp.45~51 (1993)
- S. M. George, "A principal ideal theorem analougue for modules over commutative rings", Comm. Algebra, 22(6), pp.2083~2099 (1994)
- A. Haghany, "Endoprime modules", Acta Math. Hungar, 106(1-2), pp.89~99 (2005)
- L. Levy, "Torsion-free and divisible modules over non-integral domains", Canad. J. Math, 15, pp.132~151 (1963)
- C. Lomp, "Prime elements in partially ordered groupoids applied to modules and Hopf algebra actions", J. Algebra Appl, 4(1), pp.77~97 (2005)
- J. C. McConnell, "Non-commutative Noetherian Rings" (1987), Wiley-Interscience
- P. F. Smith, "Essentially compressible modules and rings", J. Algebra, 304(2), pp.812~831 (2006)
- P. F. Smith, "Submodules of direct sums of compressible modules", Comm. Algebra, 36(8), pp.3042~3049 (2008)
- Y. Tolooei, "On rings whose modules have nonzero homomorphisms to nonzero submodules", Publ. Mat, 57(1), pp.107~122 (2013)
- R. Wisbauer, "Modules and Algebras: Bimodule Structure and Group Action on Algebras" (1996), Addison-Wesley-Longman
- H. P. Yu, "On quasi-duo rings", Glasgow Math. J, 37(1), pp.21~31 (1995)
- J. Zelmanowitz, "Weakly semisimple modules and density theory", Comm. Algebra, 21(5), pp.1785~1808 (1993)