Localization (Hungerford, Section 3.4)

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1 Lemma. (1) If p is an ideal which is maximal with respect to the property that p is not finitely generated then p is prime. (2) If p is an ideal maximal with respect to the property that p is not principal, then p is prime. Theorem. (1) If every prime ideal in a commutative ring R is finitely generated, then R is noetherian. (2) If every prime ideal in an integral domain R is principal then R is a principal ideal domain. Localization (Hungerford, Section 3.4) Concept. Additive functor. Definition. The localization functor with respect to a multiplicative set in the center of a ring and the natural transformation θ : S 1. Concept. The total quotient ring for a commutative ring and the quotient field of an integral domain. Proposition. The map θ : S 1 is an isomorphism if and only if for all s S, the map m sm is an automorphism. Universal Characterization. θ S 1 P

2 2 Proposition. For given S, localization w.r.t. S is a functor and θ is natural, i. e. θ ϕ N θ N S 1 S 1 ϕ S 1 N. Proposition. The localization functor is exact and additive. Corollary. If N is a submodule of an R-module and S is a multiplicative set in the center of R, thens 1 N can be identified as a submodule of S 1,and S 1 /S 1 N S 1 (/N). If I is a (two-sided) ideal in R and S = {s + I s S} R/I,thenS 1 I is an ideal in S 1 R and the rings S 1 R/S 1 I and S 1 (R/I) are isomorphic. Proposition. If R is noetherian then S 1 R is noetherian. (note: We need not assume that R is commutative. As always, though, we require that S Center R.) Proposition. If and N are R-modules and S is a multiplicative set in the center of R, then Hom S 1 R(S 1, S 1 N) Hom R (S 1, S 1 N). Localization and Associated Primes for Commutative Rings Note. An ideal a in a commutative ring R is prime if and only if the set of elements of R not in a forms a multiplicative set in R. Lemma [Hungerford, Theorem 2.2, p. 378]. If S is a multiplicative set in a commutative ring R and a is an ideal such that a S = then there exists at least one ideal p maximal with respect to the properties p a and p S =. Furthermore, any such ideal is prime.

3 3 Consequence [Hungerford, Theorem 2.6, p. 379]. Let a be an ideal in R and let r R. The following are equivalent: (1) r belongs to every prime ideal which contains a. (2) r does not belong to any multiplicative set S such that S a =. (3) For some positive integer k, r k a. Theorem. The set of nilpotent elements in a commutative ring forms an ideal. This is called the nil radical of the ring and it equals the intersection of all the prime ideals of the ring. (The nil radical itself is usually not prime.) Notation. If p is a prime ideal in a commutative ring R and S = R p, thenwe write p = S 1. Proposition [Hungerford, Theorem 4.10, p. 146]. The prime ideals of S 1 R are in one-to-one correspondence with the prime ideals of R which are disjoint from S. Corollary. If p is a prime ideal in R then R p is a local ring and pr p = J(R p ). Nakayama s Lemma [Hungerford, Lemma 4.5, p. 388]. If R is a not necessarily commutative ring and J its Jacobson radical and if is an R-module and N a submodule such that = N + J,thenN=. Lemma. Let m and let and only if S ann m =. ann m = {r R rm =0}. Then m/1 0 S 1 if Localization-Globalization Theorem. Let, N, P be modules over a commutative ring R. (1) If m 1,m 2,thenm 1 =m 2 if and only if m 1 /1=m 2 /1 m for all maximal ideals m. (2) =0 ifandonlyif m = 0 for all maximal ideals m. (3) Suppose that N, P.ThenN=Pif and only if N m = P m for all maximal ideals m. (4) If ϕ Hom R (,N) thenϕis monic [epic] if and only if ϕ m : m N m is monic [epic] for all maximal ideals m. (5) A sequence N P is exact if and only if the induced sequence m N m P m is exact for all maximal ideals m.

4 4 Warning. m N m for all maximal ideals m does not, in general, imply that N. Corollary: Chinese Remainder Theorem. Let a 1,...,a n be ideals in a commutative ring R such that a i + a j = R for all i j. Let be an R-module. Then a i n 1 a i. Theorem. Hilbert Basis Theorem [Hungerford, Theorem 4.9, p. 391.] Definition. We say that a prime ideal p is an associated prime for if there exists m such that p =annm. We write Ass (sometimes called the assassinator of ) for the set of associated primes for. A module is called p-primary if Ass = {p}. (Somewhat inconsistently, a submodule N of is called a p-primary submodule if /N is p-primary. C.f. Hungerford, top of p Note that Hungerford assumes that the modules are finitely generated.) Lemma. A prime ideal p is an associated prime for if and only if contains a submodule isomorphic to R/p. Proposition. If is an R-module and p is a prime ideal, the following conditions are equivalent: (1) is p-primary. (2) The natural map θ : p is monic and ( r p)( m )( k 1) r k m =0. Proposition. If p is a prime ideal then R/p is p-primary. Proposition. If p is maximal in the family of ideals {annm m }, then p is prime. Consequently if R is noetherian then Ass = =0.

5 5 Proposition. If N then AssN Ass AssN Ass/N. Definition. Supp = {p p 0}. Note. By the Localization-Globalization Theorem, =0 Supp =. Proposition. If is finitely generated then Supp = {p p is prime and p ann }. Example. Let R = Z and = Q/Z. Then 0 is a prime ideal and ann = 0 but 0 / Supp. Proposition. Ass Supp. Conversely, if p is minimal among the prime ideals in Supp then p Ass. Proposition. Supp S 1 R S 1 = { ps 1 R p Supp & p S = } Ass S 1 R S 1 = { ps 1 R p Supp & p S = } Ass R S 1 = { ps 1 R p Supp & p S = }. Proposition. If is a finitely generated module over a commutative noetherian ring then Ass is a finite set. (note: In general, Supp will not be finite.) Definition. We say that r R is a zero divisor on a module if rm =0for some m 0. Important Lemma [Hungerford, Theorem 2.3, p. 378]. If p 1,...,p n are prime ideals in a commutative ring R and a is an ideal such that a n 1 p i,thena p i for some i. Proposition. If R is noetherian then {p p Ass } is the set of elements in R which are zero divisors in. Corollary. If R is noetherian then {p p Ass R} equals the set of zero divisors in R.

6 6 odules with Finite Length over a Commutative Noetherian Ring Definition and Proposition [Hungerford, pp ]. A module over a ring R is said to have finite length if and only if it has a composition series, and in this case we define length to be the length of this composition series. The Jordan-Holder Theorem asserts that length is independent of the particular composition series. A module has finite length if and only if it is both noetherian and artinian. Proposition. If is an artinian module then Ass consists of maximal ideals. Proposition. If is a module such that Ass consists of maximal ideals, then Ass = Supp and for every p Ass, the canonical map p is a surjection and p {m ( k)p k m=0}. Theorem. A module over a commutative noetherian ring has finite length if and only if it is finitely generated and all its associated primes are maximal. Proposition. If is a module with finite length over a commutative noetherian ring R then Ass is finite and the canonical maps p for p Ass = Supp induce an isomorphism p. Ass Corollary. A commutative noetherian ring is artinian if and only if every prime ideal is maximal (including the zero ideal, if applicable). If this is the case, then R is a finite product of local rings each of which has a unique prime ideal. Theorem. Let R be a commutative noetherian local ring and let m be its unique maximal prime ideal. The following conditions are equivalent: (1) R is artinian. (2) m is the only prime ideal in R. (3) Ass R = {m}. (4) m k = 0 for some positive integer k. (5) The injective envelope of R/m is finite generated. (6) There exists a finitely generated injective R-module.

7 7 Injective odules over Commutative Noetherian Rings Theorem. If p is a prime ideal in a commutative noetherian ring R, then the injective envelope E of R/p is indecomposable. oreover, every indecomposable injective R-module is isomorphic to the injective envelope of R/p for some p. Furthermore E E p and Ass E = {p}. Corollary. Every injective module over a commutative noetherian ring R is a (possibly infinite) direct sum of indecomposable injective modules. Note. This is never true over a non-noetherian ring.