| Summary: | Let <inline-formula><math display="inline"><semantics><mi>Γ</mi></semantics></math></inline-formula> be a commutative monoid, <inline-formula><math display="inline"><semantics><mrow><mi>R</mi><mo>=</mo><msub><mo>⨁</mo><mrow><mi>α</mi><mo>∈</mo><mi>Γ</mi></mrow></msub><msub><mi>R</mi><mi>α</mi></msub></mrow></semantics></math></inline-formula> a <inline-formula><math display="inline"><semantics><mi>Γ</mi></semantics></math></inline-formula>-graded ring and <i>S</i> a multiplicative subset of <inline-formula><math display="inline"><semantics><msub><mi>R</mi><mn>0</mn></msub></semantics></math></inline-formula>. We define <i>R</i> to be a graded <i>S</i>-Noetherian ring if every homogeneous ideal of <i>R</i> is <i>S</i>-finite. In this paper, we characterize when the ring <i>R</i> is a graded <i>S</i>-Noetherian ring. As a special case, we also determine when the semigroup ring is a graded <i>S</i>-Noetherian ring. Finally, we give an example of a graded <i>S</i>-Noetherian ring which is not an <i>S</i>-Noetherian ring.
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