Math 171: Abstract Algebra, Fall 2014- Assignment 10
1. Let I be an ideal of a commutative ring R with an identity. The radical of I is defined to be rad(I) = {r ∈ R | rn ∈ I for some n ∈ Z≥1}.
(a) Prove rad(I) is an ideal of R that contains I. Give an example to show that the containment can be proper.
(b) An ideal I of R is said to be radical if rad(I) = I. Prove that every maximal ideal of a ring R is a radical ideal.
2. Let S be a commutative ring with an identity. Let R be a subring of S that has the same mulitiplicative identity as S, and let a ∈ S\R. We define the ring R[a] to be the smallest subring of S containing a and R. Prove that
R[a] = {r0 + r1a + r2a2 + · · · + rnan| ri ∈ R, n ∈ Z≥0}.
3. Throughout, we let F be a field.
(a) Prove that every ideal in F[x] is a principal ideal.
(b) Let f(x) ∈ F[x], and consider the ideal I = (f(x)). Prove that if f(x) is an irreducible polynomial in F[x] (meaning that if f(x) = p(x)q(x) in F[x] then either p(x) or q(x) is a constant, i.e. is in F), then F[x]/I is a field.
(c) Construct a field of size 125.
4. A Principal Ideal Domain (abbreviated PID) is an integral domain in which every ideal is principal. Suppose R is a PID and a, b ∈ R are non-zero. Then there exists d ∈ R such that (d) = (a, b).
(a) Prove that if d' ∈ R with d'|a and d'|b then d'|d (here, x|y means there exists r ∈ R such that y = rx).
(b) Prove d is unique up to multiplication by a unit in R.
5. Let S ⊆ C[x1, x2, . . . , xn]. We define the affine variety of S, denoted V(S), to be the subset of Cn given by
V(S) := {x ∈ Cn| f(x) = 0 for all f ∈ S}
We say X ⊆ Cn is an affine variety if X = V(S) for some S ∈ C[x1, x2, . . . , xn].
(a) Let S = {x21 - x22, x1 - x22} in C[x1, x2]. Determine V(S).
(b) Suppose X = V(S) (where S ⊆ C[x1, x2, . . . , xn] is an affine variety. Let I = (S), the ideal generated by S in C[x1, x2, . . . , xn]. Prove X = V(I).
(c) If I, J are ideals of C[x1, x2, . . . , xn] and I ⊆ J, what is the relationship between V(I) and V(J)? Prove your claims.
(d) From part (b), every variety in Cn is of the form V(I) for some ideal I of C[x1, x2, . . . , xn]. Suppose X and Y are varieties in Cn, with X = V(I) and Y = V(J). Show that X ∩ Y and X ∪ Y are varieties as well by expressing each as V(I) for some ideal I of C[x1, x2, . . . , xn].
(e) A variety X ⊆ Cn is said to be reducible if there exist subsets X1, X2 ⊆ X such that both X1, X2 are varieties, X1, X2 ∉ {∅, X}, and X = X1 ∪ X2. Draw two different reducible varieties. Explain.
(f) Determine all ideals of the ring C[x] (Hint: Every ideal in C[x] is a principal ideal (why?)). Use this to determine all varieties in C. What are the varieties of the maximal ideals in C[x]?