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`https://hdl.handle.net/10216/25790`

Author(s): | Christian Lomp A.J. Pena |

Title: | A note on prime modules |

Issue Date: | 2000 |

Abstract: | In this note we compare some notions of primeness for modules existing in the literature. We characterize the prime left R-modules such that the left annihilator of every element is a (two-sided) ideal of R, where R is an associative ring with unity, and we prove that if M is such a left R-module then M is strongly prime. These two notions are studied by Beachy [B75]. Furthermore, if M is projectiveas left R=(0 : M)-module then M is B-prime in the sense of Bican et al. [BJKN]. On the other hand, if M is faithful then M is (strongly) prime if and only if M is strongly prime (or an SP-module) in the sense of Handelman-Lawrence [HL] if and only if M is torsionfree and R is a domain. In particular this happens if R is commutative. |

Description: | In this note we compare some notions of primeness for modules existing in the literature. We characterize the prime left R-modules such that the left annihilator of every element is a (two-sided) ideal of R, where R is an associative ring with unity, and we prove that if M is such a left R-module then M is strongly prime. These two notions are studied by Beachy [B75]. Furthermore, if M is projectiveas left R=(0 : M)-module then M is B-prime in the sense of Bican et al. [BJKN]. On the other hand, if M is faithful then M is (strongly) prime if and only if M is strongly prime (or an SP-module) in the sense of Handelman-Lawrence [HL] if and only if M is torsionfree and R is a domain. In particular this happens if R is commutative. |

Subject: | Álgebra, Matemática Algebra, Mathematics |

Scientific areas: | Ciências exactas e naturais::Matemática Natural sciences::Mathematics |

URI: | https://repositorio-aberto.up.pt/handle/10216/25790 |

Document Type: | Artigo em Revista Científica Internacional |

Rights: | openAccess |

License: | https://creativecommons.org/licenses/by-nc/4.0/ |

Appears in Collections: | FCUP - Artigo em Revista Científica Internacional |

This item is licensed under a Creative Commons License