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Homework Statement
In the process of trying to prove something else I found it would be helpful if rn[itex]\in[/itex]I, where I is an ideal, n[itex]\in[/itex]N, and r[itex]\in[/itex]R and R is a ring, then r is in I.
Homework Equations
I is an ideal if a[itex]\in[/itex]I and b[itex]\in[/itex]I then a+b[itex]\in[/itex]I, a[itex]\in[/itex]I and r[itex]\in[/itex]R then ar[itex]\in[/itex]I, and I is not the empty set.
The Attempt at a Solution
Base Case: Assume r1[itex]\in[/itex]I. Then r[itex]\in[/itex]I.
Inductive Case: Assume that if rn[itex]\in[/itex]I then r[itex]\in[/itex]I for all n<n+1.
Assume rn+1[itex]\in[/itex]I. Since rn+1= rn*r then either rn or r is in I. We only need to show the first case works since the second is trivial. If rn[itex]\in[/itex]I then r[itex]\in[/itex]I by the inductive hypothesis. (QED)Is this correct?
thank you for your time.