Some results on simple complete ideals having one characteristic pair
<p>Let <em>α</em> be a regular local two-dimensional ring, and let<span style="text-decoration: underline;"> </span><em><span style="text-decoration: underline;">m </span>= (x, y) </em>be its maximal ideal. Let <em>m >...
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| Format: | Article |
| Language: | English |
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Università degli Studi di Catania
2003-05-01
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| Series: | Le Matematiche |
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| Online Access: | http://www.dmi.unict.it/ojs/index.php/lematematiche/article/view/178 |
| _version_ | 1818855474676629504 |
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| author | Silvio Greco Karlheinz Kiyek |
| author_facet | Silvio Greco Karlheinz Kiyek |
| author_sort | Silvio Greco |
| collection | DOAJ |
| description | <p>Let <em>α</em> be a regular local two-dimensional ring, and let<span style="text-decoration: underline;"> </span><em><span style="text-decoration: underline;">m </span>= (x, y) </em>be its maximal ideal. Let <em>m > n > 1</em> be coprime integers, and let<span style="text-decoration: underline;"><em> p</em></span> be the integral closure of the ideal<em> (x^m , y^n )</em>. Then<span style="text-decoration: underline;"> <em>p</em></span> is a simple complete <span style="text-decoration: underline;"><em>m</em></span>-primary ideal, and its value semigroup is generated by <em>m, n</em>.</p><p>We construct a minimal system of generators <em>{z_0 , . . . , z_n }</em> of <span style="text-decoration: underline;"><em>p</em></span>, and from this we get a minimal system of generators of the polar ideal <span style="text-decoration: underline;"><em>p'</em></span> of<span style="text-decoration: underline;"> <em>p</em></span>, consisting of<em> n = θ</em> elements. In particular, we show that <span style="text-decoration: underline;"><em>p</em></span> and <span style="text-decoration: underline;"><em>p'</em></span> are monomial ideals. When <em>α = κ[ [ x, y ] ]</em>, a ring of formal power series over an algebraically closed field <em>κ</em> of characteristic zero, this implies the existence of some relevant property.</p> |
| first_indexed | 2024-12-19T08:09:11Z |
| format | Article |
| id | doaj.art-9c31bc6ed2ee4b0b99dd077fa5750adf |
| institution | Directory Open Access Journal |
| issn | 0373-3505 2037-5298 |
| language | English |
| last_indexed | 2024-12-19T08:09:11Z |
| publishDate | 2003-05-01 |
| publisher | Università degli Studi di Catania |
| record_format | Article |
| series | Le Matematiche |
| spelling | doaj.art-9c31bc6ed2ee4b0b99dd077fa5750adf2022-12-21T20:29:40ZengUniversità degli Studi di CataniaLe Matematiche0373-35052037-52982003-05-01581333156Some results on simple complete ideals having one characteristic pairSilvio GrecoKarlheinz Kiyek<p>Let <em>α</em> be a regular local two-dimensional ring, and let<span style="text-decoration: underline;"> </span><em><span style="text-decoration: underline;">m </span>= (x, y) </em>be its maximal ideal. Let <em>m > n > 1</em> be coprime integers, and let<span style="text-decoration: underline;"><em> p</em></span> be the integral closure of the ideal<em> (x^m , y^n )</em>. Then<span style="text-decoration: underline;"> <em>p</em></span> is a simple complete <span style="text-decoration: underline;"><em>m</em></span>-primary ideal, and its value semigroup is generated by <em>m, n</em>.</p><p>We construct a minimal system of generators <em>{z_0 , . . . , z_n }</em> of <span style="text-decoration: underline;"><em>p</em></span>, and from this we get a minimal system of generators of the polar ideal <span style="text-decoration: underline;"><em>p'</em></span> of<span style="text-decoration: underline;"> <em>p</em></span>, consisting of<em> n = θ</em> elements. In particular, we show that <span style="text-decoration: underline;"><em>p</em></span> and <span style="text-decoration: underline;"><em>p'</em></span> are monomial ideals. When <em>α = κ[ [ x, y ] ]</em>, a ring of formal power series over an algebraically closed field <em>κ</em> of characteristic zero, this implies the existence of some relevant property.</p>http://www.dmi.unict.it/ojs/index.php/lematematiche/article/view/178Simple complete idealsPolar idealValue semigroupTwo- dimensional regular local ringsValuation associated to a simple idealMinimal system of generatorsMonomial ideals |
| spellingShingle | Silvio Greco Karlheinz Kiyek Some results on simple complete ideals having one characteristic pair Le Matematiche Simple complete ideals Polar ideal Value semigroup Two- dimensional regular local rings Valuation associated to a simple ideal Minimal system of generators Monomial ideals |
| title | Some results on simple complete ideals having one characteristic pair |
| title_full | Some results on simple complete ideals having one characteristic pair |
| title_fullStr | Some results on simple complete ideals having one characteristic pair |
| title_full_unstemmed | Some results on simple complete ideals having one characteristic pair |
| title_short | Some results on simple complete ideals having one characteristic pair |
| title_sort | some results on simple complete ideals having one characteristic pair |
| topic | Simple complete ideals Polar ideal Value semigroup Two- dimensional regular local rings Valuation associated to a simple ideal Minimal system of generators Monomial ideals |
| url | http://www.dmi.unict.it/ojs/index.php/lematematiche/article/view/178 |
| work_keys_str_mv | AT silviogreco someresultsonsimplecompleteidealshavingonecharacteristicpair AT karlheinzkiyek someresultsonsimplecompleteidealshavingonecharacteristicpair |