# Answer to Question #22759 in Abstract Algebra for sanches

Question #22759

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.

Expert's answer

For instance, the 2-dimensionalnoetherian domain

ideals: (

*R*= Z[[*x*]] is not*J*-semisimple by factthat J(R)=J(Z)+xZ[[x]] = xZ[[x]] – nonzero, but has infinitely many maximalideals: (

*p, x*) for*p*= 2*,*3*,*5*, . . . .*
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