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A note on prime modules
| Summary: | 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 projective as 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. |
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| Subject: | Algebra, Mathematics Álgebra, Matemática |
| Country: | Portugal |
| Document type: | journal article |
| Access type: | Open |
| Associated institution: | Repositório Aberto da Universidade do Porto |
| Language: | English |
| Origin: | Repositório Aberto da Universidade do Porto |
| _version_ | 1850560649400156160 |
|---|---|
| conditionsOfAccess_str | open access |
| country_str | PT |
| 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 projective as 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. |
| documentTypeURL_str | http://purl.org/coar/resource_type/c_6501 |
| documentType_str | journal article |
| id | a5cb82c3-29fb-4cdf-bd4d-5ab2618597f7 |
| identifierHandle_str | https://hdl.handle.net/10216/25790 |
| language | eng |
| relatedInstitutions_str_mv | Repositório Aberto da Universidade do Porto |
| resourceName_str | Repositório Aberto da Universidade do Porto |
| spellingShingle | A note on prime modules Algebra, Mathematics Álgebra, Matemática |
| title | A note on prime modules |
| topic | Algebra, Mathematics Álgebra, Matemática |