Quasi-one-fibered and projectively full ideals in 2-dimensional Muhly rational singularities
\((R, m)\) is assumed to be a 2-dimensional Muhly rational singularity, a class that contains a 2-dimensional regular local rings. The main result is as follows (in order to simplify the formulation \( m\) is assumed to be the quasi-one-fibered ideal associated with ord\(_R\): Let \(I\) be a complete \( m\)-primary ideal of \(R\), let \(T(I)=\{v_i,\dots, v_n\}\) and let \(I_i\) be the quasi-one-fibered ideal associated with \(v_i\), \(i=1,\dots,n\). Then there exist natural numbers \(s, u_1,\dots,u_n\), such that Im\(^s=I_1^{u_1}\cdot\cdot\cdot I_n^{u_n}\). We mention another noteworthy result: If \((R, m)\) is a 2-dimensional Muhly rational singularity, then \(R\) is regular, if and only if, then for every complete \( m\)-primary ideal \(I\) of \(R\), there exist positive integers \(e_i\), such that \(I=A_{v_1}^{e_1}\cdot\cdot\cdot A_{v_n}^{e_n}\). Here the ideals \(A_{v_i}\) are the one-fibered ideals associated with the Rees valuation of \(I\)).
scientific article; zbMATH DE number 6673769
scientific article