FOURTH PROBLEM SHEET FOR COMMUTATIVE ALGEBRA
M4P55
AMBRUS PÁL
not assessed
1. Let R be a ring. Prove that the following are equivalent:
(i) R has exactly one proper prime ideal.
(ii) Every element of R is either a unit or nilpotent.
(iii) The quotient ring R/N (R) is a field.
Definition 0.1. Two ideals a, b of a ring R are coprime if a + b = R.
Now let R be a ring and let a1 , a2 , . . . , an be ideals of R. Consider the homomorphism:
n
Y
(R/ai )
φ : R −→
i=1
given by the rule x 7→ (x + a1 , x + a2 , · · · , x + an ).
2. Show that
Q
T
(i) if ai and aj are coprime whenever i 6= j, then ai = ai ,
(ii) the map φ is surjective if and only if T
ai and aj are coprime whenever i 6= j,
(iii) the map φ is injective if and only if ai = (0).
3. Show that dim C[x] = 1, dim Z = 1 and dim Z[x] = 2, where dim denotes the
Krull dimension.
4. Let S be a multiplicatively closed subset of R and let M be a finitely generated
R-module. Show that
S −1 AnnR (M ) = AnnS −1 R (S −1 M ).
5. For every subset A of R = C[x1 , x2 , . . . , xn ] we define:
V (A) = {(α1 , . . . , αn ) ∈ Cn |f (α1 , . . . , αn ) = 0
(∀f ∈ A)}.
Show that
T
S
(a) we have j∈J V (aj ) = V ( j∈J aj ) for every set {aj }j∈J of ideals of R,
(b) we have V (a1 ) ∪ V (a2 ) = V (a1 a2 ) for every a1 , a2 / R,
(c) we have V (R) = ∅ and V ({0}) = Cn .
Definition 0.2. Recall that a topological space is an ordered pair X = (X, C) where
X is a set and C is a collection of subsets of X satisfying the following axioms:
(i) the empty set and X are in C,
(ii) the system C is closed under arbitrary intersection,
(iii) the system C is closed under finite union.
Date: November 27, 2016.
1
2
AMBRUS PÁL
The elements of C are called the closed sets of X. The data C is called a topology on
the set X. By the above the sets of the form V (I) form a topology on Cn , which
is called the Zariski-topology. Its closed subsets are by definition the Zariski-closed
subsets of Cn .