TY - JOUR
T1 - Multiplication modules and homogeneous idealization II
AU - Ali, Majid M.
PY - 2007
Y1 - 2007
N2 - All rings are commutative with identity and all modules are unital. Let R be a ring, M an R-module and R(M), the idealization of M. Homogenous ideals of R(M) have the form I(+)N, where I is an ideal of R and N a submodule of M such that IM ⊆ N. A ring R (M) is called a homogeneous ring if every ideal of R (M) is homogeneous. In this paper we continue our recent work on the idealization of multiplication modules and give necessary and sufficient conditions for a homogeneous ideal to be an almost (generalized, weak) multiplication, projective, finitely generated flat, pure or invertible (q-invertible). We determine when a ring R(M) is a general ZPI-ring, distributive ring, quasi-valuation ring, P-ring, coherent ring or finite conductor ring. We also introduce the concept of weakly prime submodules generalizing weakly prime ideals. Various properties and characterizations of weakly prime submodules of faithful multiplication modules are considered.
AB - All rings are commutative with identity and all modules are unital. Let R be a ring, M an R-module and R(M), the idealization of M. Homogenous ideals of R(M) have the form I(+)N, where I is an ideal of R and N a submodule of M such that IM ⊆ N. A ring R (M) is called a homogeneous ring if every ideal of R (M) is homogeneous. In this paper we continue our recent work on the idealization of multiplication modules and give necessary and sufficient conditions for a homogeneous ideal to be an almost (generalized, weak) multiplication, projective, finitely generated flat, pure or invertible (q-invertible). We determine when a ring R(M) is a general ZPI-ring, distributive ring, quasi-valuation ring, P-ring, coherent ring or finite conductor ring. We also introduce the concept of weakly prime submodules generalizing weakly prime ideals. Various properties and characterizations of weakly prime submodules of faithful multiplication modules are considered.
KW - Flat module
KW - Homogeneous ring
KW - Idealization
KW - Invertible submodule
KW - Multiplication module
KW - Projective module
KW - Pure submodule
KW - Weakly prime submodule
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M3 - Article
AN - SCOPUS:41549089322
SN - 0138-4821
VL - 48
SP - 321
EP - 343
JO - Beitrage zur Algebra und Geometrie
JF - Beitrage zur Algebra und Geometrie
IS - 2
ER -