Math 515 homework #8 Due date: 19980211 D. J. Bernstein 19980204 Exercise 17. Let R be a ring. Let I be an ideal of R. Let u be an element of R that is invertible modulo I; i.e., there is an element v ∈ R for which 1 − uv ∈ I. Prove that u is invertible modulo I^{2}. (Thus u is invertible modulo I^{n} for arbitrarily large n.) Exercise 18. Let R be a ring. Let f_{1}, …, f_{m} be elements of R. Prove that (Rf_{1} + ⋯ + Rf_{m})^{e_{1}+···+e_{m}} is contained in Rf_{1}^{e_{1}} + Rf_{2}^{e_{2}} + ⋯ + Rf_{m}^{e_{m}}. Exercise 19. Eisenbud 2.20.