Commutative algebra - Department of Mathematical Sciences - old ...

14 1. A DICTIONARY ON RINGS AND IDEALS
1.3.6. Example. An ideal in Z is a prime ideal if it is generated by 0 or a prime
number. Any nonzero prime ideal is a maximal ideal.
1.3.7. Definition. For an ideal I in a ring R the radical is
√ I = {a ∈ R|a n ∈ I for some n}
a is nilpotent is a n = 0 for some positive integer n. A ring is reduced if the
nilradical √ 0 = 0, that is if 0 is the only nilpotent element.
1.3.8. Proposition. (1) The radical of an ideal is an ideal.
(2) The nilradical is contained in any prime ideal.
(3) A domain is reduced.
Proof. (1) If am , bn ∈ I then by the binomial formula
(a + b) m+n m+n
m + n
=
a
k
m+n−k b k ∈ I
k=0
and the radical is an ideal. (2) (3) Clearly a nilpotent element is contained in any
prime ideal.
1.3.9. Exercise. (1) Show that the characteristic of a domain is either 0 or a prime
number.
(2) Let m, n be a natural numbers. Show that n + (m) ∈ Z/(m) is a unit if and only if
m, n are relative prime.
(3) Let m be a natural number. Show that Z/(m) is reduced if m is square free.
(4) Show that a product of reduced rings is reduced.
(5) Let a be nilpotent. Show that 1 − a is a unit.
(6) Let I, J be ideals. Show that √ IJ = √ I ∩ J = √ I ∩ √ J.
(7) Assume an ideal I is contained in a prime ideal P . Show that √ I ⊂ P .
1.4. Chinese remainders
1.4.1. Definition. Ideals I, J ⊂ R are comaximal ideals if I + J = R.
1.4.2. Proposition (Chinese remainder theorem). Let I1, . . . , Ik be pairwise comaximal
ideals in a ring R. Then
(1) For a1, . . . , ak ∈ R there is a a ∈ R, such that a−am ∈ Im for m = 1, . . . , k
(2)
(3) The product of projections
is an isomorphism.
Proof. (1) For each m
I1 · · · Ik = I1 ∩ · · · ∩ Ik
R/I1 · · · Ik → R/I1 × · · · × R/Ik
R =
(Im + In) = Im +
n=m
n=m
So choose um ∈ Im and vm ∈
n=m In with um + vm = 1. Put a = a1v1 + · · · +
akvk. Then a − am = · · · + amum + · · · ∈ Im. (2) For a in the intersection assume
by induction that a ∈ I2 · · · Ik. From the proof of (1) a = u1a + av1 ∈ I1 · · · Ik.
(3) Subjectivity follows from (1). The kernel is the intersection which by (2) is the
product. 1.2.9 gives the isomorphism.
In