 Here "k""p" denotes the image of "k" under the Frobenius morphism "a"→"a""p".Here is an example of a ring that is a G-ring but not a J-2 ring and so not quasi-excellent.If "R" is the subring of the polynomial ring "k"["x"1,"x"2...] in infinitely many generators generated by the squares and cubes of all generators, and "S" is obtained from "R" by adjoining inverses to all elements not in any of the ideals generated by some "x""n", then "S" is a 1-dimensional Noetherian domain that is not a J-1 ring as "S" has a cusp singularity at every closed point, so the set of singular points is not closed, though it is a G-ring.