TY - JOUR
T1 - Some Remarks on Multiplication and Projective Modules II
AU - Ali, Majid M.
PY - 2013/1
Y1 - 2013/1
N2 - All rings are commutative with identity and all modules are unital. Let R be a ring and M an R-module. In our recent work [6] we investigated faithful multiplication modules and the properties they have in common with projective modules. In this article, we continue our study and investigate faithful multiplication and locally cyclic projective modules and give several properties for them. If M is either faithful multiplication or locally cyclic projective then M is locally either zero or isomorphic to R. We show that, if M is a faithful multiplication module or a locally cyclic projective module, then for every submodule N of M there exists a unique ideal Γ(N) ⊆ Tr(M) such that N = Γ(N)M. We use this result to show that the structure of submodules of a faithful multplication or locally cyclic projective module and their traces are closely related. We also use the trace of locally cyclic projective modules to study their endomorphisms.
AB - All rings are commutative with identity and all modules are unital. Let R be a ring and M an R-module. In our recent work [6] we investigated faithful multiplication modules and the properties they have in common with projective modules. In this article, we continue our study and investigate faithful multiplication and locally cyclic projective modules and give several properties for them. If M is either faithful multiplication or locally cyclic projective then M is locally either zero or isomorphic to R. We show that, if M is a faithful multiplication module or a locally cyclic projective module, then for every submodule N of M there exists a unique ideal Γ(N) ⊆ Tr(M) such that N = Γ(N)M. We use this result to show that the structure of submodules of a faithful multplication or locally cyclic projective module and their traces are closely related. We also use the trace of locally cyclic projective modules to study their endomorphisms.
KW - Locally cyclic projective module
KW - Multiplication module
KW - Ring of endormophisms
KW - Trace of a module
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U2 - 10.1080/00927872.2011.628724
DO - 10.1080/00927872.2011.628724
M3 - Article
AN - SCOPUS:84872416859
SN - 0092-7872
VL - 41
SP - 195
EP - 214
JO - Communications in Algebra
JF - Communications in Algebra
IS - 1
ER -