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

7.3. ARTINIAN RINGS 89
Proof. (1) Let P1, . . . , Pn be the prime and maximal ideals 7.3.11. The nilradical
√ 0 = P1 ∩ · · · ∩ Pn by 5.1.7. Conclude by Chinese remainders 1.4.3. (2) √ 0 k =
√ 0 k+1 for some k. If √ 0 k = 0 let (a) be minimal among ideals I such that
I √ 0 k = 0. By minimality (a) = (a) √ 0 k , so a = ab for some b ∈ √ 0 k . But b is
nilpotent 1.3.8, so a = 0 gives a contradiction. It follows, that √ 0 k = 0.
7.3.13. Proposition. A ring R is artinian if and only if it has finite length.
Proof. The factor module √ 0 i / √ 0 i+1 is an artinian module over R/ √ 0 which is
a product of fields, so it has finite length.
7.3.14. Corollary. Let R be an artinian ring and M a module. The following are
equivalent.
(1) M is finite.
(2) M has finite length.
(3) M is finite presented.
7.3.15. Corollary. Let R be artinian and R → S a finite ring homomorphism.
(1) S is artinian.
(2) A finite length S-module N is by restriction of scalars a finite length Rmodule.
(3) A finite length R-module M gives by change of rings M ⊗R S as finite length
S-module.
7.3.16. Proposition. Let M be a R-module of finite length.
(1) The ring R/ Ann(M) is artinian.
(2) There are only finitely many prime ideals Ann(M) ⊂ P .
(3) Any prime ideal Ann(M) ⊂ P is maximal.
(4) M is finite presented.
Proof. (1) Let x ∈ M, then R/ Ann(x) Rx is artinian. If x1, . . . , xn generate
M, then Ann(M) = Ann(x1)∩· · ·∩Ann(Xn). By 7.3.8 R/ Ann(M) is artinian.
7.3.17. Example. Let K be a field. The ring R =
artinian.
(1) The maximal ideals in R are
(2) The simple types are
N
Pi = {a : N → K|a(i) = 0}
R/Pi Ji = {a : N → K|a(j) = 0, i = j}
(3) Ji are the only simple ideals and the sum is an ideal
Ji = J = R
(4) J is an ideal which has no complement in R.
i
K = {a : N → K} is not
7.3.18. Exercise. (1) Show that a vector space is artinian if and only if it is finite dimensional.
(2) Show that Z is not artinian.
(3) Show that R[X] is not artinian for a nonzero ring R.
(4) Show that Q[X]/(X 2 − X) is artinian.