When Are Graded Rings Graded <i<S</i<-Noetherian Rings

Let <inline-formula<<math display="inline"<<semantics<<mi mathvariant="sans-serif"<Γ</mi<</semantics<</math<</inline-formula< be a commutative monoid, <inline-formula<<math display="inline"<<semantics<<...
Ausführliche Beschreibung

Let <inline-formula<<math display="inline"<<semantics<<mi mathvariant="sans-serif"<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 mathvariant="sans-serif"<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 mathvariant="sans-serif"<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. Ausführliche Beschreibung