Math 514 homework #9 Due date: 19950925 D. J. Bernstein 19950918 Let R be a subring of a field F. Then F is an R-module under F ’s multiplication. A sub-R-module of F is a submodule of F closed under multiplication by R. Exercise 24. Prove that xR is a sub-R-module of F for any x ∈ F. Exercise 25. Prove that (xR)(yR) = (xy)R. Exercise 26. Prove that xR ÷ yR = (x/y)R for y ≠ 0. Exercise 27. Let I be a sub-R-module of F, and let M be any nonzero module in F. Prove that the module product IM and module ratio I ÷ M are sub-R-modules of F. An R-submodule I of F is called a fractional ideal over R if (1) I is nonzero, i.e., I has a nonzero element;

(2) dI ⊆ R for some nonzero d ∈ R. Exercise 28. Prove that (a/b)R is a fractional R-ideal for any nonzero a, b ∈ R. Exercise 29. Let I and J be fractional R-ideals. Prove that IJ and I ÷ J are fractional R-ideals. A fractional R-ideal M is invertible over R if MN = R for some fractional ideal N. Recall that if MN = R then N = R ÷ M. Exercise 30. Prove that aR is invertible for any nonzero a ∈ R. Exercise 31. Let I be a fractional R-ideal, and let M be an invertible fractional R-ideal. Prove that I ÷ M = I(R ÷ M). Exercise 32. Let M and I be invertible fractional R-ideals. Prove that MI is invertible.