| Summary: | Let <i>R</i> be an associative ring with unity, <i>X</i> be a finite group, <i>H</i> be a subgroup of <i>X</i>, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">G</mi></semantics></math></inline-formula> be a set of left coset representatives for the left action of <i>H</i> on <i>X</i>. In this article, we introduce two different ways to put <i>R</i> into a non-trivial <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">G</mi></semantics></math></inline-formula>-weak graded ring that is a ring graded by the set <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">G</mi></semantics></math></inline-formula> which is defined with a binary operation ∗ and satisfying an algebraic structure with specific properties. The first one is by choosing a subset <i>S</i> of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">G</mi></semantics></math></inline-formula> such that <i>S</i> is a group under the ∗ operation and putting <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>R</mi><mi>t</mi></msub><mo>=</mo><mn>0</mn></mrow></semantics></math></inline-formula> for all <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><mo>∈</mo><mi mathvariant="fraktur">G</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>t</mi><mo>∉</mo><mi>S</mi></mrow></semantics></math></inline-formula>. The second way, which is the most important, is induced by combining the operation ∗ defined on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">G</mi></semantics></math></inline-formula> and the coaction <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mo mathsize="70%">◁</mo></semantics></math></inline-formula> of <i>H</i> on <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="fraktur">G</mi></semantics></math></inline-formula>. Many examples are provided.
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