### Abstract

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.

Original language | English |
---|---|

Pages (from-to) | 321-343 |

Number of pages | 23 |

Journal | Beitrage zur Algebra und Geometrie |

Volume | 48 |

Issue number | 2 |

Publication status | Published - 2007 |

### Fingerprint

### Keywords

- Flat module
- Homogeneous ring
- Idealization
- Invertible submodule
- Multiplication module
- Projective module
- Pure submodule
- Weakly prime submodule

### ASJC Scopus subject areas

- Algebra and Number Theory
- Geometry and Topology

### Cite this

*Beitrage zur Algebra und Geometrie*,

*48*(2), 321-343.

**Multiplication modules and homogeneous idealization II.** / Ali, Majid M.

Research output: Contribution to journal › Article

*Beitrage zur Algebra und Geometrie*, vol. 48, no. 2, pp. 321-343.

}

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

UR - http://www.scopus.com/inward/record.url?scp=41549089322&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=41549089322&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:41549089322

VL - 48

SP - 321

EP - 343

JO - Beitrage zur Algebra und Geometrie

JF - Beitrage zur Algebra und Geometrie

SN - 0138-4821

IS - 2

ER -