1. Extension-Contraction and the Spectrum of a Polynomial Ring

∗ ∗ ∗ Extension and Contraction of Ideals ∗ ∗ ∗

If R ⊆ S are commutative rings then every ideal I of R defines an ideal of S by expansion: I^{e} = IS, and every ideal J of S defines an ideal of R by contraction J^{e} = J ∩ R. More generally, if f: R → S is a ring-homomorphism, then every I ⊴ R defines an ideal I^{c} = f(I)S of S and every J ⊴ S an ideal J^{e} = f^{−1}(J) of R. (When f is understood, I^{e} is often written as IS and J^{e} as J ∩ R by abuse of notation even if f is not injective.)

Contraction preserves the property of being a prime ideal (or a primary ideal); extension, in general, does not.

Clearly, for ideals A, B of R and C, D of S, A ⊆ B ⇒ A^{c} ⊆ B and C ⊆ D ⇒ C^{e} ⊆ D^{e}; and A^{ce} ⊆ A and D^{ec} ⊇ D.

Therefore, for every ring homomorphism f: R → S, extension and contraction give a bijective correspondence between ideals of R of the form J^{c} = J ∩ R (J an ideal of S) and ideals of S of the form I^{e} = IS (I an ideal of R). In two noteable cases, all ideals of S are of the form I^{e}:

• Let H be an ideal of R and π:R→R/H, π(r)=r+H, the canonical projection. Then extension and contraction induce a bijection between all ideals of R/H and the ideals of R containing H. In both directions of this bijection the properties of being a prime ideal, a primary ideal, or a G-ideal are preserved. Note that for a subset S of R/H π^{−1}(S) = ⋃_{r+H∈S} r + H.

• Let S be a multiplicative subset of R, S^{−1}R the ring of quotients with denomiators in S and f: R → S^{−1}R the canonical map f(r) = r/1 (if R has a unit element, or f(r) = rs/s otherwise). Recall that f is injective whenever S contains no zero-divisors. Then extension and contraction induce a bijection between all ideals of S^{−1}R and those ideals of R that are disjoint from S. Most importantly, if S = R ⊆ P, P a prime ideal, then the bijection is between all ideals of R_{P} and the ideals of R contained in P. Recall that for I an ideal of R, the rings R[x]/I[x] and (R/I)[x] are canonically

isomorphic via a_{0} +a_{1}x+…+a_{n}x^{n} +I[x] ↦ (a_{0} +I)+(a_{1} +I)x+…+(a_{n} +I)x^{n}.

∗ ∗ ∗ Spectrum of a Polynomial Ring ∗ ∗ ∗

Let R be a polynomial ring and P a prime ideal of R. Then every prime ideal Q of R[x] with Q ∩ R ⊇ P contains P [x] and is therefore of the form π^{−1}(Q̄), where π: R[x] → R[x]/P [x] = (R/P)[x] is the canonical projection and Q̄ is a prime ideal of (R/P)[x]. Those Q among them with Q ∩ R = P correspond to those Q̄ for which Q̄ ∩ (R/P) = (0 + P).

Let D be a domain with quotient field K. Then every prime ideal Q of D[x] with Q ∩ D = (0) is of the form Q̄ ∩ D[x], where Q̄ is a prime ideal of K[x] (and thus Q̄ = (f), where f is an irreducible polynomial of K[x].

To summarize: to determine for a given prime ideal P of R all prime ideals Q of R[x] with Q ∩ R = P: consider the polynomial ring K_{P} [x], where K_{P} is the quotient field of R/P, i.e. K_{P} = (R/P \{0+P })^{−1}R/P for each monic irreducible f ∈ K_{P} [x], consider all polynomials of R/P [x] that are divisible by f in K_{P} [x], i.e. all products fg with g ∈ K_{P} [x], such that fg ∈ R/P [x]. Then the collection of all polynomials of D[x] whose residue class (under the projection of coefficients from R to R/P is of the form fg (for some g ∈ K_{P} [x]) is a prime ideal of D[x] and all prime ideals Q of D[x] with Q ∩ R = P are of this form.