Answer to Question #25251 in Abstract Algebra for Mohammad

Question

Show that from if I is a nil ideal in any ring R, then I[t] ⊆ rad R[t], follows rad (R[t]) = (Nil*R)[t] for any ring R.

Answer 1

The inclusion “⊆” in conclusion follows fromAmitsur’s Theorem. The reverse inclusion “⊇” follows from given assumption: " I[t]⊆rad R[t], for I is a nil ideal ".So, rad (R[t]) = (Nil*R)[t].