Math 514 homework #15 Due date: 19951016 D. J. Bernstein 19951009 The set of units of a ring R is denoted R^{∗}. A ring is local if it has exactly one maximal ideal. Exercise 47. Let R be a local ring with maximal ideal P. Prove that R − P = R^{∗}. Exercise 48. Let R be a local ring with maximal ideal P. Let M be a finitely generated R-module such that M = P M. Prove that M = 0. In the following two exercises R is a ring, x is an element of R, and I is an ideal of R satisfying I = xR + I^{2}. Exercise 49. Prove that I = xR + I^{n} for all n ≥ 2. Exercise 50. Prove that I^{n} = x^{n}R + I^{n+1} for all n ≥ 1.