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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 xnhas 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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