189-235A: Basic Algebra I Assignment 8

Due: Wednesday, November 10

1. Let F be a field, and let f : F ⟶ R be a homomorphism satisfying f(0) ≠ f(1). Show that f is necessarily injective, and that the image of f is isomorphic to F.

2. Show that the image of Z under any homomorphism is isomorphic either to Z or to a finite ring.

Let R be a commutative ring and let I be an ideal of R. Prove or disprove the following statements.

3. If R is an integral domain, then so is R/I.

4. If every ideal in R is principal, the same is true of R/I.

5. If every ideal in R/I is principal, the same is true of R.

6. Let R = Z[x], and let I be the ideal (p, x^{2} + 1) generated by the integer prime p and the polynomial x^{2} + 1. Show that R/I is isomorphic to Z/5Z × Z/5Z is p = 5, and is isomorphic to a field with 49 elements if p = 7.

Extra credit.

7. Let F be a field and let R = F [[x]] denote the ring of formal power series with coefficients in F, i.e., the set of expressions of the form

∑_{n=0}^{∞} a_{n}x^{n}, a_{n} ∈ F,

where the addition and multiplication are performed by formally expanding out the sums and products (without worrying about issues of convergence, which don’t make sense in an arbitrary field F anyways!)

Let I = (x) be the ideal generated by the power series x. Show that R/I is isomorphic to F. Show that any element of R which does not belong to I is invertible. Conclude that any non-trivial ideal of R is contained in I. (A ring with this property is called a local ring, a terminology arising from the prototypical example F [[x]], because power series can be thought of as “functions defined only around the value x = 0”.)