6.7. NOETHERIAN RINGS311Proposition 6.7.2.For a ringR(not necessarily commutative, not neces-sarily with identity), the following are equivalent:(a)Every ideal ofRis finitely generated.(b)Rsatisfies the ascending chain condition for ideals.Proof.Suppose that every ideal ofRis finitely generated. LetI1I2I3be an infinite, weakly increasing sequence of ideals ofR. ThenID [nInis an ideal ofR, so there exists a finite setSsuch thatIis theideal generated byS. Each element ofSis contained in someIn; sincetheInare increasing, there exists anNsuch thatSIN. SinceIis thesmallest ideal containingS, we haveIIN[nInDI. It follows thatInDINfor allnN. This shows thatRsatisfies the ACC for ideals.Suppose thatRhas an idealIthat is not finitely generated. Then forany finite subsetSofI, the ideal generated bySis properly contained inI. Hence there exists an elementf2In.S/, and the ideal generatedbySis properly contained in that generated byS[ ffg. An inductive
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