On z-Ideals and z ◦ -Ideals of Power Series Rings
Let R be a commutative ring with identity and R[[x]] be the ring of formal power series with coefficients in R. In this article we consider sufficient conditions in order that P[[x]] is a minimal prime ideal of R[[x]] for every minimal prime ideal P of R and also every minimal prime ideal of R[[...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
Islamic Azad University
2013-06-01
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| Series: | Journal of Mathematical Extension |
| Online Access: | http://ijmex.com/index.php/ijmex/article/view/200/122 |
| _version_ | 1819208889798754304 |
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| author | A. Rezaei Aliabad R. Mohamadian |
| author_facet | A. Rezaei Aliabad R. Mohamadian |
| author_sort | A. Rezaei Aliabad |
| collection | DOAJ |
| description | Let R be a commutative ring with identity and R[[x]] be
the ring of formal power series with coefficients in R. In this article
we consider sufficient conditions in order that P[[x]] is a minimal prime
ideal of R[[x]] for every minimal prime ideal P of R and also every
minimal prime ideal of R[[x]] has the form P[[x]] for some minimal
prime ideal P of R. We show that a reduced ring R is a Noetherian
ring if and only if every ideal of R[[x]] is nicely-contractible (we call an
ideal I of R[[x]] a nicely-contractible ideal if (I ∩ R)[[x]] ⊆ I). We will
trivially see that an ideal I of R[[x]] is a z-ideal if and only if we have
I = (I, x) in which I is a z-ideal of R and also we show that whenever
every minimal prime ideal of R[[x]] is nicely-contractible, then I[[x]] is
a z
◦
-ideal of R[[x]] if and only if I is an ℵ0-z
◦
-ideal |
| first_indexed | 2024-12-23T05:46:34Z |
| format | Article |
| id | doaj.art-d2208c30dff64f65aa59a806a8cf1c31 |
| institution | Directory Open Access Journal |
| issn | 1735-8299 1735-8299 |
| language | English |
| last_indexed | 2024-12-23T05:46:34Z |
| publishDate | 2013-06-01 |
| publisher | Islamic Azad University |
| record_format | Article |
| series | Journal of Mathematical Extension |
| spelling | doaj.art-d2208c30dff64f65aa59a806a8cf1c312022-12-21T17:58:04ZengIslamic Azad UniversityJournal of Mathematical Extension1735-82991735-82992013-06-017293108On z-Ideals and z ◦ -Ideals of Power Series RingsA. Rezaei Aliabad0R. Mohamadian1Chamran UniversityChamran UniversityLet R be a commutative ring with identity and R[[x]] be the ring of formal power series with coefficients in R. In this article we consider sufficient conditions in order that P[[x]] is a minimal prime ideal of R[[x]] for every minimal prime ideal P of R and also every minimal prime ideal of R[[x]] has the form P[[x]] for some minimal prime ideal P of R. We show that a reduced ring R is a Noetherian ring if and only if every ideal of R[[x]] is nicely-contractible (we call an ideal I of R[[x]] a nicely-contractible ideal if (I ∩ R)[[x]] ⊆ I). We will trivially see that an ideal I of R[[x]] is a z-ideal if and only if we have I = (I, x) in which I is a z-ideal of R and also we show that whenever every minimal prime ideal of R[[x]] is nicely-contractible, then I[[x]] is a z ◦ -ideal of R[[x]] if and only if I is an ℵ0-z ◦ -idealhttp://ijmex.com/index.php/ijmex/article/view/200/122 |
| spellingShingle | A. Rezaei Aliabad R. Mohamadian On z-Ideals and z ◦ -Ideals of Power Series Rings Journal of Mathematical Extension |
| title | On z-Ideals and z ◦ -Ideals of Power Series Rings |
| title_full | On z-Ideals and z ◦ -Ideals of Power Series Rings |
| title_fullStr | On z-Ideals and z ◦ -Ideals of Power Series Rings |
| title_full_unstemmed | On z-Ideals and z ◦ -Ideals of Power Series Rings |
| title_short | On z-Ideals and z ◦ -Ideals of Power Series Rings |
| title_sort | on z ideals and z ◦ ideals of power series rings |
| url | http://ijmex.com/index.php/ijmex/article/view/200/122 |
| work_keys_str_mv | AT arezaeialiabad onzidealsandzidealsofpowerseriesrings AT rmohamadian onzidealsandzidealsofpowerseriesrings |