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## Pure Submodules of Multiplication Modules

Citations: | 3 - 1 self |

### Citations

641 |
Commutative ring theory
- Matsumura
- 1986
(Show Context)
Citation Context ...dules of flat modules are flat, from which it follows that pure ideals are flat ideals [13]. Any direct summand N of an R-module M is pure, while the converse is true if M/N is of finite presentation =-=[16]-=-. It is proved [9, Corollaries 11.21 and 11.23] that if M is a flat module then for any submodule N of M the following are equivalent: M. M. Ali, D. J. Smith: Pure Submodules of Multiplication Modules... |

216 | Multiplicative Ideal Theory - Gilmer - 1968 |

125 | Lectures on Modules and Rings, - Lam - 1999 |

49 |
Algebra: rings, modules and categories
- Faith
- 1973
(Show Context)
Citation Context ...ated faithful multiplication modules are torsionless (Corollaries 2.3 and 2.4). All rings are commutative with identity and all modules are unital. For the basic concepts used, we refer the reader to =-=[9]-=-–[13] and [19]. 1. Pure submodules Let R be a ring andM an R-module. Cohn [8] called a submodule N ofM a pure submodule if the sequence 0 → N ⊗ E → M ⊗ E is exact for every R-module E. Anderson and Fu... |

43 | Multiplicative Theory of Ideals - Larsen, McCarthy - 1971 |

41 |
Prüfer Domains
- Fontana, Huckaba, et al.
- 1997
(Show Context)
Citation Context ...∈ R such that a = axa, and hence R is von Neumann regular. Projective modules are characterized as direct summand of free modules. Hence they are pure submodules of free modules. The trace ideal (see =-=[10]-=-) of a module M is defined as T (M) = ∑ f∈Hom(M,R) f(M). It is known [22],[23] that if M is a projective R-module, then (1) M = T (M)M . (2) annM = ann(T (M)). (3) T (M) is a pure ideal. We return to ... |

35 |
On the free product of associative rings
- COHN
- 1959
(Show Context)
Citation Context ...). All rings are commutative with identity and all modules are unital. For the basic concepts used, we refer the reader to [9]–[13] and [19]. 1. Pure submodules Let R be a ring andM an R-module. Cohn =-=[8]-=- called a submodule N ofM a pure submodule if the sequence 0 → N ⊗ E → M ⊗ E is exact for every R-module E. Anderson and Fuller [6] called the submodule N a pure submodule of M if IN = N ∩ IM for ever... |

34 |
Modules and Rings,
- Kasch
- 1982
(Show Context)
Citation Context ... faithful multiplication modules are torsionless (Corollaries 2.3 and 2.4). All rings are commutative with identity and all modules are unital. For the basic concepts used, we refer the reader to [9]–=-=[13]-=- and [19]. 1. Pure submodules Let R be a ring andM an R-module. Cohn [8] called a submodule N ofM a pure submodule if the sequence 0 → N ⊗ E → M ⊗ E is exact for every R-module E. Anderson and Fuller ... |

25 |
Multiplication Modules,
- Barnard
- 1981
(Show Context)
Citation Context ...of a module 0. Introduction Let R be a ring and M a unital R-module. Then M is called a multiplication module if for every submodule N of M there exists an ideal I of R such that N = IM, [3], [5] and =-=[7]-=-. An ideal I of R which is a multiplication module is called a multiplication ideal. Let R be a ring and K and L be submodules of an R-module M . The residual of K by L is [K : L], the set of all x in... |

22 |
Algebraic Numbers,
- Ribenboim
- 1972
(Show Context)
Citation Context ... multiplication modules are torsionless (Corollaries 2.3 and 2.4). All rings are commutative with identity and all modules are unital. For the basic concepts used, we refer the reader to [9]–[13] and =-=[19]-=-. 1. Pure submodules Let R be a ring andM an R-module. Cohn [8] called a submodule N ofM a pure submodule if the sequence 0 → N ⊗ E → M ⊗ E is exact for every R-module E. Anderson and Fuller [6] calle... |

13 | Some remarks on multiplication modules, - Smith - 1988 |

9 |
Some remarks on multiplication ideals,
- Anderson
- 1980
(Show Context)
Citation Context ..., trace of a module 0. Introduction Let R be a ring and M a unital R-module. Then M is called a multiplication module if for every submodule N of M there exists an ideal I of R such that N = IM, [3], =-=[5]-=- and [7]. An ideal I of R which is a multiplication module is called a multiplication ideal. Let R be a ring and K and L be submodules of an R-module M . The residual of K by L is [K : L], the set of ... |

9 |
and Categories of Modules. Springer-Verlag
- Anderson, Fuller
- 1992
(Show Context)
Citation Context ... and [19]. 1. Pure submodules Let R be a ring andM an R-module. Cohn [8] called a submodule N ofM a pure submodule if the sequence 0 → N ⊗ E → M ⊗ E is exact for every R-module E. Anderson and Fuller =-=[6]-=- called the submodule N a pure submodule of M if IN = N ∩ IM for every ideal I of R. Ribenboim [19] defined N to be pure in M if rM ∩ N = rN for each r ∈ R. Although the first condition implies the se... |

8 |
On radicals of submodules of finitely generated modules
- McCasland, Moore
- 1986
(Show Context)
Citation Context ... 6= M and whenever rm ∈ P, for some m ∈ M and r ∈ R, then m ∈ P or r ∈ [P :M ]. The M -radical, rad N, of a submodule N of M is defined as the intersection of all prime submodules of M containing N , =-=[17]-=-. If I is an ideal of R, then √ I is defined as the intersection of all prime ideals of R containing I. If I is a pure (and hence idempotent) ideal of R, then I = I √ I, and if I is a finitely generat... |

5 |
On projective modules of finite rank
- Vasconcelos
- 1969
(Show Context)
Citation Context ...are characterized as direct summand of free modules. Hence they are pure submodules of free modules. The trace ideal (see [10]) of a module M is defined as T (M) = ∑ f∈Hom(M,R) f(M). It is known [22],=-=[23]-=- that if M is a projective R-module, then (1) M = T (M)M . (2) annM = ann(T (M)). (3) T (M) is a pure ideal. We return to investigate the trace of a pure submodule in Part 2, but for now we apply Theo... |

4 | Finite and infinite collections of multiplication modules, - Ali, Smith - 2001 |

4 |
Some remarks on multiplication ideals II ,
- Anderson
- 2000
(Show Context)
Citation Context ...odule, trace of a module 0. Introduction Let R be a ring and M a unital R-module. Then M is called a multiplication module if for every submodule N of M there exists an ideal I of R such that N = IM, =-=[3]-=-, [5] and [7]. An ideal I of R which is a multiplication module is called a multiplication ideal. Let R be a ring and K and L be submodules of an R-module M . The residual of K by L is [K : L], the se... |

4 |
Multiplication modules and ideals,
- Low, Smith
- 1990
(Show Context)
Citation Context ...y generated faithful multiplication R-module. Let N be a pure submodule of M. Then N is finitely generated if and only if T (N) is finitely generated. Proof. See [15, Lemma 1.4 (ii)]. Low and Smith =-=[15]-=- proved that for a faithful multiplication module M, ⋂ f∈Hom(M,R) ker f = 0, from which it follows that M is torsionless in the sense that it can be embedded in a direct product of copies of R. We can... |

4 | The residual of finitely generated multiplication modules, - Naoum, Hasan - 1986 |

3 |
Finiteness in projective ideals
- Vasconcelos
- 1973
(Show Context)
Citation Context ...ules are characterized as direct summand of free modules. Hence they are pure submodules of free modules. The trace ideal (see [10]) of a module M is defined as T (M) = ∑ f∈Hom(M,R) f(M). It is known =-=[22]-=-,[23] that if M is a projective R-module, then (1) M = T (M)M . (2) annM = ann(T (M)). (3) T (M) is a pure ideal. We return to investigate the trace of a pure submodule in Part 2, but for now we apply... |

2 | Projective, flat and multiplication modules, - Ali, Smith - 2002 |

2 | On the ideal equation I(B - Anderson - 1983 |

2 | Projective ideals of finite type - Smith - 1969 |