Ring-theoretic properties of commutative algebras of invariants.
The authors introduce classes of rings that generalize the class of affine or noetherian rings. Let \(\mathcal C\) be a class of rings which has the property that if \(C\in {\mathcal C}\) then every prime homomorphic image of \(C\) is in \(\mathcal C\). A ring \(C\) is nearly \(\mathcal C\) of level \(0\) iff \(C\in {\mathcal C}\). Inductively \(C\) is nearly \(\mathcal C\) of level \(t>0\) if \(C\) has a common ideal \(A\not= 0\) with a nearly \(\mathcal C\) ring \(R\) of level \(<t\), such that every prime homomorphic image of \(R\) is nearly \(\mathcal C\) of level \(<t\), and every homomorphic image of \(C/A\) is nearly \(\mathcal C\) of level \(<t\). It is shown that if \(C\) is a nearly Noetherian ring, then \(C\) satisfies the ascending chain condition and descending chain condition on prime ideals. It is further shown that nearly affine algebras are catenary, satisfy the height formula and the Krull dimension equals the transcendence degree of the quotient field. Algebras of invariants of Lie super-algebras studied by \textit{I. Kanto} and \textit{I. Trisham} [Commun. Algebra 25, 2039-2070 (1997; Zbl 0889.17001)] are nearly affine. In the last section it is shown that if \(C\) is a nearly Dedekind domain with respect to a maximal \(A\triangleleft C\), then every ideal \(I\) of \(C\) can be written in the form \(I=P_1^{i_1}\cdots P_m^{i_m}I'\), where the \(P_j\) are distinct maximal ideals \(\not=A\) of \(C\), and either \(I'=C\) (the case \(I\not\subseteq A\)) or \(I'\triangleleft C\) is primary with \(\sqrt{I'}=A\).
scientific article; zbMATH DE number 1967910
scientific article