Question

Let R be a commutative domain that is not a field. Show that not always R is not J-semisimple implies R is semilocal, if R is a noetherian domain.

Answer 1

For instance, the 2-dimensionalnoetherian domain R = Z[[x]] is not J-semisimple by factthat J(R)=J(Z)+xZ[[x]] = xZ[[x]] – nonzero, but has infinitely many maximal
ideals: (p, x) for p = 2, 3, 5, . . . .