### 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. Homogeneous ideals of R (M) have the form I_{(+}N where I is an ideal of R, N a submodule of M and IM ⊆ N. The purpose of this paper is to investigate how properties of a homogeneous ideal I_{(+)}N of R (M) are related to those of I and N. We show that if M is a multiplication R-module and I_{(-)})N is a meet principal (join principal) homogeneous ideal of R (M) then these properties can be transferred to I and N. We give some conditions under which the converse is true. We also show that I_{(-)}N is large (small) if and only if N is large in M (I is a small ideal of R).

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

Pages (from-to) | 249-270 |

Number of pages | 22 |

Journal | Beitrage zur Algebra und Geometrie |

Volume | 47 |

Issue number | 1 |

Publication status | Published - 2006 |

### Fingerprint

### Keywords

- Idealization
- Join principal submodule
- Large submodule
- Meet principal module
- Multiplication module
- Small submodule

### ASJC Scopus subject areas

- Algebra and Number Theory
- Geometry and Topology

### Cite this

*Beitrage zur Algebra und Geometrie*,

*47*(1), 249-270.

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

Research output: Contribution to journal › Article

*Beitrage zur Algebra und Geometrie*, vol. 47, no. 1, pp. 249-270.

}

TY - JOUR

T1 - Multiplication modules and homogeneous idealization

AU - Ali, Majid M.

PY - 2006

Y1 - 2006

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. Homogeneous ideals of R (M) have the form I(+N where I is an ideal of R, N a submodule of M and IM ⊆ N. The purpose of this paper is to investigate how properties of a homogeneous ideal I(+)N of R (M) are related to those of I and N. We show that if M is a multiplication R-module and I(-))N is a meet principal (join principal) homogeneous ideal of R (M) then these properties can be transferred to I and N. We give some conditions under which the converse is true. We also show that I(-)N is large (small) if and only if N is large in M (I is a small ideal of R).

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. Homogeneous ideals of R (M) have the form I(+N where I is an ideal of R, N a submodule of M and IM ⊆ N. The purpose of this paper is to investigate how properties of a homogeneous ideal I(+)N of R (M) are related to those of I and N. We show that if M is a multiplication R-module and I(-))N is a meet principal (join principal) homogeneous ideal of R (M) then these properties can be transferred to I and N. We give some conditions under which the converse is true. We also show that I(-)N is large (small) if and only if N is large in M (I is a small ideal of R).

KW - Idealization

KW - Join principal submodule

KW - Large submodule

KW - Meet principal module

KW - Multiplication module

KW - Small submodule

UR - http://www.scopus.com/inward/record.url?scp=34748886446&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=34748886446&partnerID=8YFLogxK

M3 - Article

VL - 47

SP - 249

EP - 270

JO - Beitrage zur Algebra und Geometrie

JF - Beitrage zur Algebra und Geometrie

SN - 0138-4821

IS - 1

ER -