Math 515 homework #5 Due date: 19980204 D. J. Bernstein 19980128 Exercise 11. Eisenbud 1.2. (“Prime” is an abbreviation for “prime ideal.” The notation (I, f) means the smallest ideal containing I ∪ {f}, namely I + fR.)

Localization of modules. Let M be a module over a ring R. Let U be a multiplicative subset of R, i.e., a commutative monoid under 1, ·. Consider the following relation on M ×U: (m, u) ∼ (m′, u′) if xum′ = xu′m for some x ∈ U. Exercise 12. Prove that ∼ is an equivalence relation. The equivalence class of (m, u) under ∼ is written m/u. The localization of M at U, written U^{−1}M or M[U^{−1}], is the set of all equivalence classes. Addition of fractions is the binary operation (m, u), (m′, u′) ↦ (u′m + um′, uu′) on M × U. Exercise 13. Prove that addition of fractions induces a binary operation on M[U^{−1}]. This binary operation on M[U^{−1}] is commutative and associative, since addition of fractions is commutative and associative. It has a neutral element, namely 0/1, and inverses, namely m/u ↦ (−m)/u, so it makes M[U^{−1}] into a commutative group. The map m ↦ m/1 is a group homomorphism from M to M[U^{−1}].