Prove that $R^{(k)} \cap R^{(n)} = R^{((k-1,n-1)+1)}$.

Question

Let R be a ring and $k,n \in N \setminus\{1\}$. And let $R^{(k)} = \{a \in R | a^{k}=a \}$.

Prove that $R^{(k)} \cap R^{(n)} = R^{((k-1,n-1)+1)}$.

Please give me ideals how to prove it.

Answer 1

Hint $\,\ (a^k\!-a,\,a^n\!-a) = a(a^{\large k-1}-1,\,a^{\large n-1}-1) = a(a^{(\large k-1,n-1)}-1)\ $ by here