Question

Let R ⊆ S be rings. Show that R ∩ Nil*(S) ⊆ Nil*(R).

Answer 1

Let r ∈ R ∩ Nil*(S). Consider any m-systemM of the ring R such that r ∈ M. By definition, M is also an m-systemof S, so 0 ∈ M since r ∈Nil*(S). It follows that r ∈Nil*(R).