Mappings between the lattices of saturated submodules with respect to a prime ideal

Abstract

Let $\mathfrak{S}_p(_RM)$ be the lattice of all saturated submodules of an $R$-module $M$ with respect to a prime ideal $p$ of a commutative ring $R$. We examine the properties of the mappings $\eta:\mathfrak{S}_p(_RR)\rightarrow \mathfrak{S}_p(_RM)$ defined by $\eta(I)=S_p(IM)$ and $\theta:\mathfrak{S}_p(_RM)\rightarrow \mathfrak{S}_p(_RR)$ defined by $\theta(N)=(N:M)$, in particular considering when these mappings are lattice homomorphisms. It is proved that if $M$ is a semisimple module or a projective module, then $\eta$ is a lattice homomorphism. Also, if $M$ is a faithful multiplication $R$-module, then $\eta$ is a lattice epimorphism. In particular, if $M$ is a finitely generated faithful multiplication $R$-module, then $\eta$ is a lattice isomorphism and its inverse is $\theta$. It is shown that if $M$ is a distributive module over a semisimple ring $R$, then the lattice $\mathfrak{S}_p(_RM)$ forms a Boolean algebra and $\eta$ is a Boolean algebra homomorphism.

Keywords

• Saturated submodule with respect to a prime ideal
• $\eta$-module
• $\theta$-module
• $\mathfrak{S}$-distributive module
• semisimple ring

References

1. [1] M. Alkan and Y. Tiras, On invertible and dense submodules, Comm. Algebra, 32 (10), 3911–3919, 2004.
2. [2] M. Alkan and Y. Tiras, On prime submodules, Rocky Mountain J. Math. 37 (3), 709–722, 2007.
3. [3] M.F. Atiyah and I.G. Macdonald, Introduction to Commutative Algebra, Addison- Wesley, London, 1969.
4. [4] A. Barnard, Multiplication modules, J. Algebra, 71 (1), 174–178, 1981.
5. [5] S. Burris and H.P. Sankappanavar, A Course in Universal Algebra, Springer-Verlag, New York, 1981.
6. [6] J. Dauns, Prime submodules, J. Reine Angew. Math. 298, 156–181, 1978.
7. [7] Z.A. El-Bast and P.F. Smith, Multiplication modules, Comm. Algebra, 16 (4), 755– 799, 1988.
8. [8] V. Erdogdu, Multiplication modules which are distributive, J. Pure Appl. Algebra, 54, 209–213, 1988.

9. [9] J.B. Harehdashti and H.F. Moghimi, Complete homomorphisms between the lattices of radical submodules, Math. Rep. 20(70) (2), 187–200, 2018.
10. [10] T.W. Hungerford, Algebra, Springer-Verlag, New York, 1974.
11. [11] J. Jenkins and P.F. Smith, On the prime radical of a module over a commutative ring, Comm. Algebra, 20 (12), 3593–3602, 1992.
12. [12] T.Y. Lam, A First Course in Noncommutative Rings, Springer-Verlag, New York, 1991.
13. [13] M.D. Larsen and P.J. McCarthy, Multiplicative Theory of Ideals, Academic Press, New York, 1971.
14. [14] C.P. Lu, M-radical of submodules in modules. Math. Japonica, 34 (2), 211–219, 1989.
15. [15] C.P. Lu, Saturations of submodules, Comm. Algebra, 31 (6), 2655–2673, 2003.
16. [16] C.P. Lu, A module whose prime spectrum has the surjective natural map, Houston J. Math. 33 (1), 125–143, 2007.
17. [17] R.L. McCasland and M.E. Moore, On radicals of submodules, Comm. Algebra, 19 (5), 1327–1341, 1991.
18. [18] R.L. McCasland and M.E. Moore, Prime submodules, Comm. Algebra, 20 (6), 1803– 1817, 1992.
19. [19] R.L. McCasland, M.E. Moore and P.F. Smith, On the spectrum of a Module over a commutative ring, Comm. Algebra, 25 (1), 79–103, 1997.
20. [20] H.F. Moghimi and J.B. Harehdashti, Mappings between lattices of radical submodules, Int. Electron. J. Algebra, 19, 35–48, 2016.
21. [21] P.F. Smith, Some remarks on multiplication modules, Arch. Math. 50, 223–235, 1988.
22. [22] P.F. Smith, Mappings between module lattices, Int. Electron. J. Algebra, 15, 173–195, 2014.
23. [23] P.F. Smith, Complete homomorphisms between module lattices, Int. Electron. J. Al- gebra, 16, 16–31, 2014.
24. [24] P.F. Smith, Anti-homomorphisms between module lattices, J. Commut. Algebra, 7, 567–591, 2015.