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

Morteza NOFERESTİ; Hosein FAZAELİ MOGHİMİ; Mohammad Hossein HOSSEİNİ

doi:https://doi.org/10.15672/hujms.605105

Makale detayları

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

Yazarlar:

Morteza NOFERESTİ | Hosein FAZAELİ MOGHİMİ | Mohammad Hossein HOSSEİNİ

Yayıncı:

Hacettepe University

Ana Başlıklar:

Matematik

Dergi:

Hacettepe Journal of Mathematics and Statistics 2021 Vol.50 Issue 1, pp.243-254

Belge Tipi:

Makale

Yayın Tarihi:

2/2021

Belge Dili:

İngilizce

Document ID (Org No.):

2000450816

Sıklık:

İki Aylık

eISSN:

2651-477X

pISSN:

2651-477X

DOI:

https://doi.org/10.15672/hujms.605105

Özet:

 

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

Makale detayları

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

Yazarlar: Morteza NOFERESTİ | Hosein FAZAELİ MOGHİMİ | Mohammad Hossein HOSSEİNİ

Yayıncı: Hacettepe University

Ana Başlıklar: Matematik

Dergi: Hacettepe Journal of Mathematics and Statistics 2021 Vol.50 Issue 1, pp.243-254

Belge Tipi: Makale

Yayın Tarihi: 2/2021

Belge Dili: İngilizce

Document ID (Org No.): 2000450816

Sıklık: İki Aylık

eISSN: 2651-477X

pISSN: 2651-477X

DOI: https://doi.org/10.15672/hujms.605105

Sayı

Yazarlar:

Morteza NOFERESTİ | Hosein FAZAELİ MOGHİMİ | Mohammad Hossein HOSSEİNİ

Yayıncı:

Hacettepe University

Ana Başlıklar:

Matematik

Dergi:

Hacettepe Journal of Mathematics and Statistics 2021 Vol.50 Issue 1, pp.243-254

Belge Tipi:

Makale

Yayın Tarihi:

2/2021

Belge Dili:

İngilizce

Document ID (Org No.):

2000450816

Sıklık:

İki Aylık

eISSN:

2651-477X

pISSN:

2651-477X

DOI:

https://doi.org/10.15672/hujms.605105