Math 515 homework #4 Due date: 19980130 D. J. Bernstein 19980123

Binomial coefficients. Let R be a ring. By induction (x + y)^{n} = ∑_{0≤k≤n} \binom{n}{k}x^{n−k}y^{k} in R[x][y], where \binom{n}{k} is the integer n(n − 1)(n − 2) ⋯ (n − k + 1)/k!.

Taylor’s formula. Define D : R[x] → R[x] by D(h) = h′. Write D^{k} for the kth iterate of D; for example, D^{k}(x^{n}) = n(n − 1)(n − 2) ⋯ (n − k + 1)x^{n−k}. In the following exercise f ∈ R[x]. Consider f(x + y) ∈ R[x][y]. Write f(x + y) as g_{0} + yg_{1} + y^{2}g_{2} + ⋯ with g_{k} ∈ R[x]. For example, if f = 2x^{2} + 1, then g_{0} = 2x^{2} + 1, g_{1} = 4x, g_{2} = 2, g_{3} = 0, etc. Exercise 9. Prove that k!g_{k} = D^{k}(f). Thus f(x+y) = f +yf′+· · ·. This provides an alternate proof of Exercise 2: fg+y(fg)′+⋯ = (fg)(x+y) = f(x + y)g(x + y) = (f + yf′ + ⋯)(g + yg′ + ⋯) = fg + y(f′g + g′f) + · · ·.

Modules. Let M be a commutative group. Let R be a ring. A function φ : R → M^{M} is a ring action of R on M if it satisfies the following identities: φ(r)(m + n) = φ(r)(m) + φ(r)(n); φ(r + s)(m) = φ(r)(m) + φ(s)(m); φ(1)(m) = m; and φ(rs)(m) = φ(r)(φ(s)(m)). Usually φ(r)(m) is abbreviated rm. Alternate terminology: M is a ring module over R under φ; M is a module over R under φ; M is an R-module under φ; M is an R-module. A module over a field is called a vector space. Any commutative group is a Z-module in exactly one way: the action is b ↦ multiplication by b. Any ideal of a ring R is an R-module under r ↦ (m ↦ rm).

Algebras versus modules. Let A be an R-algebra, i.e., a ring with a ring homomorphism h : R → A. The natural action of R on A is the function r ↦ (a ↦ h(r)a). Under this action, A is an R-module. For example, if I is an ideal of R then R/I is an R-module. If φ is a ring action then Im φ is a ring. Here are its operations: 0 is m ↦ 0; 1 is m ↦ m; − is f ↦ (m ↦ −f(m)); + is (f, g) ↦ (m ↦ f(m) + g(m)); · is (f, g) ↦ (m ↦ f(g(m))). The action φ is a ring homomorphism from R to Im φ; thus Im φ is an R-algebra under φ. Exercise 10. Let R and S be rings. Let h : R → S be a surjective ring homomorphism. Let φ be the natural action of R on S. Prove that Im φ and S are isomorphic R-algebras.