# Answer to Question #17660 in Abstract Algebra for Tsit Lam

Question #17660

Let R be the commutative Q-algebra generated by x1, x2, . . . with the relations (x_n)^n= 0 for all n. Show that R does not have a largest nilpotent ideal.

Expert's answer

It is not hard to show that each

*xn*has index of nilpotency exactly equal to*n*in*R*. Let*I*beany nilpotent ideal in*R*; say*In*= 0. Then*x_n*+1*is notin I*, and so*I*+ (*x_n*+1)*R*is a nilpotent ideallarger than*I*.
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