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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| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-05-01
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| Series: | Axioms |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2075-1680/14/5/365 |
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| Summary: | 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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| ISSN: | 2075-1680 |