Mathematics 893 Problem Set 8

Suggested reading: Chapter 9, sections 1 and 3. Chapter 10, sections 1-3. Section 10.5, pages 378 to 385. Section 12.1.

Solutions due Friday November 25.

(a) Page 356 # 4, 5.

(b) Page 356 #7

(c) Let k be a field and let R = k[x, y, z, w]/(xy − zw). Show that R is an integral domain. Prove that, in R, x is irreducible but not prime.

(d) Let k be a field and let R = k[x, y]/(x^{2}y). Show that R is not an integral domain. Prove that in R the ideals (y) and (y(1 − x)) are equal, but that there is no unit λ of R such that y = λy(1 − x). Possible approach: Show that k[x] and k[y] are quotient rings of R in a natural way. If λ is a unit in R what can one say about the canonical images of λ in k[x] and k[y]? What does this tell us about λ? This is in contast to the integral domain situation, Proposition 3, page 274.

(e) Let R be as in Page 358 #27. Find elements a, b, c, d in R such that ac+bd = 1, ca = 1, cb = 0, da = 0, db = 1. Let A = (a, b) and

B = ( \begin{matrix}c \\ d \end{matrix} ). Calculate the matrix products AB and BA. From

this conclude that R ≅ R^{2} as right R modules. (This is a similar conclusion to #27, phrased in the language of matrices instead of bases.)

(f) Page 220 #1 (finite rank case only). Hint: Consider the commutator subgroups, which you will have to study a bit.

Remark: We have now seen at least three variants of the notion of freeness.

(i) Free groups: Theorem 17 page 217 (ii) Polynomial rings If R is a commutative ring with 1, S is an R-

algebra, X = {x_{1}, …, x_{n}}, and ϕ : X → S is an arbitrary map of sets then there is a unique R-algebra homomorphism Φ : R[x_{1}, … x_{n}] → S such that

\xymatrix{X \ar[r]^{i} \ar[dr]_{ϕ} & R[x_{1}, …, x_{n}] \ar[d]^{Φ} \\  & S}

commutes (where i is the inclusion of X in R[x_{1} … x_{n}]). Informally this says that an R-algebra homomorphism Φ : R[x_{1}, …, x_{n}] → S is determined by the Φ(x_{i}), which can be chosen arbitrarily. For this reason Φ is often described simply by giving the Φ(x_{i}), rather than trying to write down an arbitrary element f ∈ R[x_{1} … x_{n}] and then defining Φ(f). This version ought to be somewhere in the text, but I have not yet found it.

(iii) Free modules. Theorem 6, page 354.