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

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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Bibliographic Details
Main Authors:	Azzh Saad Alshehry, Rashid Abu-Dawwas
Format:	Article
Language:	English
Published:	MDPI AG 2024-12-01
Series:	Axioms
Subjects:	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
Online Access:	https://www.mdpi.com/2075-1680/13/12/879
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Summary:	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).
ISSN:	2075-1680