Year 2014,
Volume: 15 Issue: 15, 173 - 195, 01.06.2014
Abstract
We examine the properties of certain mappings between the lattice
of ideals of a commutative ring R and the lattice of submodules of an
R-module M, in particular considering when these mappings are lattice homomorphisms.
We prove that the mapping λ from the lattice of ideals of R
to the lattice of submodules of M defined by λ(B) = BM for every ideal B
of R is a (lattice) isomorphism if and only if M is a finitely generated faithful
multiplication module. Moreover, for certain but not all rings R, there is an
isomorphism from the lattice of ideals of R to the lattice of submodules of an
R-module M if and only if the mapping λ is an isomorphism.
References
- D. D. Anderson, Some remarks on multiplication ideals, Math. Japonica, (1980), 463-469.
- D. D. Anderson and Y. Al-Shaniafi, Multiplication modules and the ideal θ(M ), Comm. Algebra, 30(7)(2002), 3383-3390.
- A. Barnard, Multiplication modules, J. Algebra, 71(1981), 174-178.
- Z. A. El-Bast and P. F. Smith, Multiplication modules, Comm. Algebra, (4)(1988), 755-779.
- R. Gilmer, Multiplicative Ideal Theory, Marcel Dekker, New York, 1972.
- J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, Wiley, Chichester, 1987.
- D. W Sharpe and P. Vamos, Injective Modules, Cambridge Tracts in Math- ematics and Mathematical Physics 62, Cambridge Univ. Press, Cambridge, S. Singh and F. Mehdi, Multiplication modules, Canad. Math. Bull., 22 (1979), 98.
- S. Singh and Y. Al-Shaniafi, Multiplication modules, Comm. Algebra, (6)(2001), 2597-2609.
- S. Singh and Y. Al-Shaniafi, A companion ideal of a multiplication module, Period. Math. Hungar., 46(1)(2003), 1-8.
- P. F. Smith, Some remarks on multiplication modules, Arch. Math., 50(1988), 422. Patrick F. Smith
- Department of Mathematics University of Glasgow Glasgow G12 8QW, Scotland UK e-mail: Patrick.Smith@glasgow.ac.uk
Year 2014,
Volume: 15 Issue: 15, 173 - 195, 01.06.2014
Abstract
References
- D. D. Anderson, Some remarks on multiplication ideals, Math. Japonica, (1980), 463-469.
- D. D. Anderson and Y. Al-Shaniafi, Multiplication modules and the ideal θ(M ), Comm. Algebra, 30(7)(2002), 3383-3390.
- A. Barnard, Multiplication modules, J. Algebra, 71(1981), 174-178.
- Z. A. El-Bast and P. F. Smith, Multiplication modules, Comm. Algebra, (4)(1988), 755-779.
- R. Gilmer, Multiplicative Ideal Theory, Marcel Dekker, New York, 1972.
- J. C. McConnell and J. C. Robson, Noncommutative Noetherian Rings, Wiley, Chichester, 1987.
- D. W Sharpe and P. Vamos, Injective Modules, Cambridge Tracts in Math- ematics and Mathematical Physics 62, Cambridge Univ. Press, Cambridge, S. Singh and F. Mehdi, Multiplication modules, Canad. Math. Bull., 22 (1979), 98.
- S. Singh and Y. Al-Shaniafi, Multiplication modules, Comm. Algebra, (6)(2001), 2597-2609.
- S. Singh and Y. Al-Shaniafi, A companion ideal of a multiplication module, Period. Math. Hungar., 46(1)(2003), 1-8.
- P. F. Smith, Some remarks on multiplication modules, Arch. Math., 50(1988), 422. Patrick F. Smith
- Department of Mathematics University of Glasgow Glasgow G12 8QW, Scotland UK e-mail: Patrick.Smith@glasgow.ac.uk
There are 11 citations in total.
Details
| Other ID | JA72VD22VT |
|---|---|
| Authors | |
| Publication Date | June 1, 2014 |
| DOI | https://doi.org/10.24330/ieja.266246 |
| IZ | https://izlik.org/JA79ED45PL |
| Published in Issue | Year 2014 Volume: 15 Issue: 15 |