The Norm Function for Commutative <inline-formula><math display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mn>2</mn></msub></semantics></math></inline-formula>-Graded Rings
Consider a commutative <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mn>2</mn></msub></semantics></math></inl...
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2024-12-01
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| author | Azzh Saad Alshehry Rashid Abu-Dawwas |
| author_facet | Azzh Saad Alshehry Rashid Abu-Dawwas |
| author_sort | Azzh Saad Alshehry |
| collection | DOAJ |
| description | Consider a commutative <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mn>2</mn></msub></semantics></math></inline-formula>-graded ring (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mo>=</mo><msub><mi>R</mi><mn>0</mn></msub><mo>⨁</mo><msub><mi>R</mi><mn>1</mn></msub></mrow></semantics></math></inline-formula>). Consequently, each element (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>R</mi></mrow></semantics></math></inline-formula>) can be uniquely expressed as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>=</mo><msub><mi>x</mi><mn>0</mn></msub><mo>+</mo><msub><mi>x</mi><mn>1</mn></msub></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub><mo>∈</mo><msub><mi>R</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>∈</mo><msub><mi>R</mi><mn>1</mn></msub></mrow></semantics></math></inline-formula>. For any <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>R</mi></mrow></semantics></math></inline-formula>, we consider the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msubsup><mi>x</mi><mn>0</mn><mn>2</mn></msubsup><mo>−</mo><msubsup><mi>x</mi><mn>1</mn><mn>2</mn></msubsup></mrow></semantics></math></inline-formula>. In this work, we examine the properties of <i>N</i> and utilize them to derive new results. Moreover, we apply this function to establish concepts such as <i>N</i>-prime ideals, <i>N</i> radicals, <i>N</i>-integral domains, and <i>N</i> fields, achieving several notable results along the way. Among our results, we demonstrate that an <i>N</i>-prime ideal is not necessarily prime. Additionally, we show that the <i>N</i> radical differs from the usual radical ideal and is not always an ideal. Furthermore, we establish that an <i>N</i>-integral domain (<i>N</i> field) is not necessarily an integral domain (field). |
| format | Article |
| id | doaj-art-36fbbca404e84054a6f4fff626611a0f |
| institution | Kabale University |
| issn | 2075-1680 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Axioms |
| spelling | doaj-art-36fbbca404e84054a6f4fff626611a0f2024-12-27T14:10:29ZengMDPI AGAxioms2075-16802024-12-01131287910.3390/axioms13120879The Norm Function for Commutative <inline-formula><math display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mn>2</mn></msub></semantics></math></inline-formula>-Graded RingsAzzh Saad Alshehry0Rashid Abu-Dawwas1Department of Mathematical Sciences, Faculty of Sciences, Princess Nourah bint Abdulrahman University, P.O. Box 84428, Riyadh 11671, Saudi ArabiaDepartment of Mathematics, Yarmouk University, Irbid 21163, JordanConsider a commutative <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mn>2</mn></msub></semantics></math></inline-formula>-graded ring (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>R</mi><mo>=</mo><msub><mi>R</mi><mn>0</mn></msub><mo>⨁</mo><msub><mi>R</mi><mn>1</mn></msub></mrow></semantics></math></inline-formula>). Consequently, each element (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>R</mi></mrow></semantics></math></inline-formula>) can be uniquely expressed as <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>=</mo><msub><mi>x</mi><mn>0</mn></msub><mo>+</mo><msub><mi>x</mi><mn>1</mn></msub></mrow></semantics></math></inline-formula>, where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>x</mi><mn>0</mn></msub><mo>∈</mo><msub><mi>R</mi><mn>0</mn></msub></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>x</mi><mn>1</mn></msub><mo>∈</mo><msub><mi>R</mi><mn>1</mn></msub></mrow></semantics></math></inline-formula>. For any <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>R</mi></mrow></semantics></math></inline-formula>, we consider the function <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mrow><mo>(</mo><mi>x</mi><mo>)</mo></mrow><mo>=</mo><msubsup><mi>x</mi><mn>0</mn><mn>2</mn></msubsup><mo>−</mo><msubsup><mi>x</mi><mn>1</mn><mn>2</mn></msubsup></mrow></semantics></math></inline-formula>. In this work, we examine the properties of <i>N</i> and utilize them to derive new results. Moreover, we apply this function to establish concepts such as <i>N</i>-prime ideals, <i>N</i> radicals, <i>N</i>-integral domains, and <i>N</i> fields, achieving several notable results along the way. Among our results, we demonstrate that an <i>N</i>-prime ideal is not necessarily prime. Additionally, we show that the <i>N</i> radical differs from the usual radical ideal and is not always an ideal. Furthermore, we establish that an <i>N</i>-integral domain (<i>N</i> field) is not necessarily an integral domain (field).https://www.mdpi.com/2075-1680/13/12/879graded ring structure<named-content content-type="equation"><inline-formula> <mml:math id="mm1001"> <mml:semantics> <mml:msub> <mml:mi mathvariant="double-struck">Z</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:semantics> </mml:math> </inline-formula></named-content>-graded ringhomogeneous element |
| spellingShingle | Azzh Saad Alshehry Rashid Abu-Dawwas The Norm Function for Commutative <inline-formula><math display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mn>2</mn></msub></semantics></math></inline-formula>-Graded Rings Axioms graded ring structure <named-content content-type="equation"><inline-formula> <mml:math id="mm1001"> <mml:semantics> <mml:msub> <mml:mi mathvariant="double-struck">Z</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:semantics> </mml:math> </inline-formula></named-content>-graded ring homogeneous element |
| title | The Norm Function for Commutative <inline-formula><math display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mn>2</mn></msub></semantics></math></inline-formula>-Graded Rings |
| title_full | The Norm Function for Commutative <inline-formula><math display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mn>2</mn></msub></semantics></math></inline-formula>-Graded Rings |
| title_fullStr | The Norm Function for Commutative <inline-formula><math display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mn>2</mn></msub></semantics></math></inline-formula>-Graded Rings |
| title_full_unstemmed | The Norm Function for Commutative <inline-formula><math display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mn>2</mn></msub></semantics></math></inline-formula>-Graded Rings |
| title_short | The Norm Function for Commutative <inline-formula><math display="inline"><semantics><msub><mi mathvariant="double-struck">Z</mi><mn>2</mn></msub></semantics></math></inline-formula>-Graded Rings |
| title_sort | norm function for commutative inline formula math display inline semantics msub mi mathvariant double struck z mi mn 2 mn msub semantics math inline formula graded rings |
| topic | graded ring structure <named-content content-type="equation"><inline-formula> <mml:math id="mm1001"> <mml:semantics> <mml:msub> <mml:mi mathvariant="double-struck">Z</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:semantics> </mml:math> </inline-formula></named-content>-graded ring homogeneous element |
| url | https://www.mdpi.com/2075-1680/13/12/879 |
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