Graded 1-Absorbing Prime Ideals over Non-Commutative Graded Rings
In this article, we define and study graded 1-absorbing prime ideals and graded weakly 1-absorbing prime ideals in non-commutative graded rings as a new class of graded ideals that lies between graded prime ideals (graded weakly prime ideals) and graded 2-absorbing ideals (graded weakly 2-absorbing...
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| author | Azzh Saad Alshehry Rashid Abu-Dawwas Rahaf Abudalo |
| author_facet | Azzh Saad Alshehry Rashid Abu-Dawwas Rahaf Abudalo |
| author_sort | Azzh Saad Alshehry |
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| description | In this article, we define and study graded 1-absorbing prime ideals and graded weakly 1-absorbing prime ideals in non-commutative graded rings as a new class of graded ideals that lies between graded prime ideals (graded weakly prime ideals) and graded 2-absorbing ideals (graded weakly 2-absorbing ideals). Let <i>G</i> be a group and let <i>R</i> be a non-commutative <i>G</i>-graded ring with nonzero unity. Let <i>P</i> be a proper graded ideal of <i>R</i>. We then say that <i>P</i> is a graded 1-absorbing prime ideal (a graded weakly 1-absorbing prime ideal) of <i>R</i> if, for each nonunit homogeneous element <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi>R</mi></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mi>R</mi><mi>s</mi><mi>R</mi><mi>t</mi><mo>⊆</mo><mi>P</mi></mrow></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><mn>0</mn><mo>}</mo><mo>≠</mo><mi>r</mi><mi>R</mi><mi>s</mi><mi>R</mi><mi>t</mi><mo>⊆</mo><mi>P</mi></mrow></semantics></math></inline-formula>), either <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mi>s</mi><mo>∈</mo><mi>P</mi></mrow></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><mo>∈</mo><mi>P</mi></mrow></semantics></math></inline-formula>. We present a number of properties and characterizations of these graded ideals. |
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| spelling | doaj-art-1d5bf2824fa94e718af359bf07add0062025-08-20T01:56:17ZengMDPI AGAxioms2075-16802025-05-0114536510.3390/axioms14050365Graded 1-Absorbing Prime Ideals over Non-Commutative Graded RingsAzzh Saad Alshehry0Rashid Abu-Dawwas1Rahaf Abudalo2Department 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, JordanDepartment of Mathematics, Yarmouk University, Irbid 21163, JordanIn this article, we define and study graded 1-absorbing prime ideals and graded weakly 1-absorbing prime ideals in non-commutative graded rings as a new class of graded ideals that lies between graded prime ideals (graded weakly prime ideals) and graded 2-absorbing ideals (graded weakly 2-absorbing ideals). Let <i>G</i> be a group and let <i>R</i> be a non-commutative <i>G</i>-graded ring with nonzero unity. Let <i>P</i> be a proper graded ideal of <i>R</i>. We then say that <i>P</i> is a graded 1-absorbing prime ideal (a graded weakly 1-absorbing prime ideal) of <i>R</i> if, for each nonunit homogeneous element <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo>∈</mo><mi>R</mi></mrow></semantics></math></inline-formula> with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mi>R</mi><mi>s</mi><mi>R</mi><mi>t</mi><mo>⊆</mo><mi>P</mi></mrow></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><mn>0</mn><mo>}</mo><mo>≠</mo><mi>r</mi><mi>R</mi><mi>s</mi><mi>R</mi><mi>t</mi><mo>⊆</mo><mi>P</mi></mrow></semantics></math></inline-formula>), either <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>r</mi><mi>s</mi><mo>∈</mo><mi>P</mi></mrow></semantics></math></inline-formula> or <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><mo>∈</mo><mi>P</mi></mrow></semantics></math></inline-formula>. We present a number of properties and characterizations of these graded ideals.https://www.mdpi.com/2075-1680/14/5/365graded prime idealsgraded weakly prime idealsgraded 1-absorbing prime idealsgraded weakly 1-absorbing prime ideals |
| spellingShingle | Azzh Saad Alshehry Rashid Abu-Dawwas Rahaf Abudalo Graded 1-Absorbing Prime Ideals over Non-Commutative Graded Rings Axioms graded prime ideals graded weakly prime ideals graded 1-absorbing prime ideals graded weakly 1-absorbing prime ideals |
| title | Graded 1-Absorbing Prime Ideals over Non-Commutative Graded Rings |
| title_full | Graded 1-Absorbing Prime Ideals over Non-Commutative Graded Rings |
| title_fullStr | Graded 1-Absorbing Prime Ideals over Non-Commutative Graded Rings |
| title_full_unstemmed | Graded 1-Absorbing Prime Ideals over Non-Commutative Graded Rings |
| title_short | Graded 1-Absorbing Prime Ideals over Non-Commutative Graded Rings |
| title_sort | graded 1 absorbing prime ideals over non commutative graded rings |
| topic | graded prime ideals graded weakly prime ideals graded 1-absorbing prime ideals graded weakly 1-absorbing prime ideals |
| url | https://www.mdpi.com/2075-1680/14/5/365 |
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