On piecewise Noetherian domains (Q2806540)

The authors are interested in studying properties of piecewise Noetherian domains. These rings were introduced in [\textit{J. A. Beachy} and \textit{W. D. Weakley}, Commun. Algebra 12, 2679--2706 (1984; Zbl 0562.13013)] with the definition as follows. A commutative ring \(R\) is piecewise Noetherian if it satisfies the ascending chain condition (ACC) on its prime ideals and has finite ideal lengths. It was shown here that the piecewise Noetherian property is inherited by factor rings and localizations and extends to integral extensions and polynomial rings. A ring is said to have Noetherian spectrum if it satisfies ACC on radical ideals. An integral domain \(R\) is said to be a piecewise \(\omega\)-Noetherian domain if (1) \(R\) satisfies ACC on prime \(\omega\)-ideals, (2) \(R\) has ACC on P-primary ideals for each prime \(\omega\)-ideal \(P\) and (3) each \(\omega\)-ideal has only finitely many prime ideals minimal over it.NEWLINENEWLINEIn the article, the authors are interested especially in studying piecewise Noetherian and piecewise \(\omega\)-Noetherian properties under similar situations as above. In particular, the major results include (1) flat overrings of piecewise Noetherian rings are piecewise Noetherian, (2) if \(R\) is a domain of finite character, the \(R\) is piecewise Noetherian if and only if \(R_M\) is piecewise Noetherian for all maximal ideals \(M\) of \(R\), (3) results on pullbacks, (4) amalgamation of \(R\) along an ideal \(J\) of \(R\), (5) a \(t\)-flat overring of a piecewise \(\omega\)-Noetherian is piecewise \(\omega\)-Noetherian, (6) if \(R\) is \(\omega\)-finite character, then \(R\) is piecewise \(\omega\)-Noetherian if and only if \(R_M\) is piecewise \(\omega\)-Noetherian for each maximal \(\omega\)-ideals \(M\) of \(R\), (7) polynomial extensions of piecewise \(\omega\)-Noetherian rings, and (8) the rings \(R=D + K[X]\) and \(R=D + K[X]]\) where \(D\) is a domain and \(K\) is the field of fractions. There are several other topics investigated as well as examples provided and an open question regarding power series expansions of piecewise Noetherian rings.