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.
Noferesti, M., Fazaeli Moghimi, H., & Hosseini, M. H. (2021). Mappings between the lattices of saturated submodules with respect to a prime ideal. Hacettepe Journal of Mathematics and Statistics, 50(1), 243-254. https://doi.org/10.15672/hujms.605105
AMA
Noferesti M, Fazaeli Moghimi H, Hosseini MH. Mappings between the lattices of saturated submodules with respect to a prime ideal. Hacettepe Journal of Mathematics and Statistics. February 2021;50(1):243-254. doi:10.15672/hujms.605105
Chicago
Noferesti, Morteza, Hosein Fazaeli Moghimi, and Mohammad Hossein Hosseini. “Mappings Between the Lattices of Saturated Submodules With Respect to a Prime Ideal”. Hacettepe Journal of Mathematics and Statistics 50, no. 1 (February 2021): 243-54. https://doi.org/10.15672/hujms.605105.
EndNote
Noferesti M, Fazaeli Moghimi H, Hosseini MH (February 1, 2021) Mappings between the lattices of saturated submodules with respect to a prime ideal. Hacettepe Journal of Mathematics and Statistics 50 1 243–254.
IEEE
M. Noferesti, H. Fazaeli Moghimi, and M. H. Hosseini, “Mappings between the lattices of saturated submodules with respect to a prime ideal”, Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 1, pp. 243–254, 2021, doi: 10.15672/hujms.605105.
ISNAD
Noferesti, Morteza et al. “Mappings Between the Lattices of Saturated Submodules With Respect to a Prime Ideal”. Hacettepe Journal of Mathematics and Statistics 50/1 (February 2021), 243-254. https://doi.org/10.15672/hujms.605105.
JAMA
Noferesti M, Fazaeli Moghimi H, Hosseini MH. Mappings between the lattices of saturated submodules with respect to a prime ideal. Hacettepe Journal of Mathematics and Statistics. 2021;50:243–254.
MLA
Noferesti, Morteza et al. “Mappings Between the Lattices of Saturated Submodules With Respect to a Prime Ideal”. Hacettepe Journal of Mathematics and Statistics, vol. 50, no. 1, 2021, pp. 243-54, doi:10.15672/hujms.605105.
Vancouver
Noferesti M, Fazaeli Moghimi H, Hosseini MH. Mappings between the lattices of saturated submodules with respect to a prime ideal. Hacettepe Journal of Mathematics and Statistics. 2021;50(1):243-54.