4. Integral elements

4.1 Definition. Let R be a commutative ring, T an R-algebra and u ∈ T. Then u is called integral over R if and only if there exists a monic polynomial f ∈ R[x] with f(u) = 0.

4.2 Definition. A ring extension R ⊆ T is called integral if every element of T is integral over R.

4.3 Theorem. Let R be a commutative ring, T an R-algebra and u ∈ T. Then the following are equivalent:

(1) u is integral over R.

(2) R[u] is finitely generated as an R-module.

(3) R[u] ⊆ A ⊆ T for some ring A that is finitely generated as an R-module.

(4) There exists a faithful R[u]-module that is finitely generated as an R module.

Proof. (1 ⇒ 2) R[u] is generated as an R-module by the powers of u. Since u is integral over R, there exists an n ∈ ℕ and r_{0}, …, r_{n−1} ∈ R such that u^{n} = r_{n−1}u^{n−1} + … + r_{1}u + r_{0}, and by induction, every power of u is expressible as an R-linear combination of 1, u, …, u^{n−1}. (2 ⇒ 3) Take A = R[u]. (3 ⇒ 4) Take A as the faithful R[u] module. (4 ⇒ 1) Let M be a faithful R[u]-module, generated by m_{1}, …, m_{n} as an Rmodule. For each w ∈ R[u], φ_{w}(m) = wm (scalar multiplication by w) is an R-module endomorphism of M, φ_{w} ∈ End_{R}(M). Let C = (c_{ij}) in M_{n}(R) be a matrix with with φ_{u}(m_{k}) = c_{1k}m_{1} + c_{2k}m_{2} + … + c_{nk}m_{n} then C represents φ_{u} ∈ End_{R}(M) with respect to the system of generators m_{1}, …, m_{n}.

By Cayley-Hamilton (3.5), χ(φ_{u}) = 0, where χ = det(xI − C) is the characteristic polynomial of C. 0 = χ(φ_{u}) = φ_{χ(u)}, means that scalar multiplication by χ(u) is the 0-mapping on M, in other words, χ(u) ∈ Ann_{R[u]}(M). Since M is a faithful R[u] module, χ(u) = 0 follows. □

4.4 Definition. Let R be a commutative ring, T an R-algebra and u ∈ T. Then u is called almost integral over R if and only if there exists a finitely generated R-submodule B of T with R[u] ⊆ B.

Comparing the definition of “almost integral” with the preceding theorem, we see that (i) “integral” implies “almost integral” (ii) for Noetherian R, the notions of “integral” and “almost integral” coincide.

(For Noetherian R, every R-submodule of a finitely generated R-module is finitely generated, such that “almost integral” implies condition (1) of 4.3.)

If R is a domain with quotient field K, then u ∈ K is almost integral over R iff there exists a non-zero d ∈ R such that du^{n} ∈ R for all n ∈ ℕ. (If u is almost intgral and R[u] ⊆ B, where b_{1}, …, b_{m} generate B as a R-module then there exists a non-zero (common denominator) d with db_{i} ∈ R (1 ≤ i ≤ m) and hence dR[u] ⊆ R; conversely, if there exists d ∈ R, d ≠ 0, with dR[u] ⊆ R then R[u] ⊆ d^{−1}R, where d^{−1}R is finitely generated, even cyclic, as an R-module.)

4.5 Definition. A domain D with quotient field K is called completely integrally closed if every element of K that is almost integral over D lies in D.

As we have seen, for Noetherian domains the notions of “integrally closed” and “completely integrally closed” coincide.

If R ⊆ T are commutative rings and T is finitely generated as an R-module, then T is also finitely generated over R as a ring, but the converse does not hold in general, viz. the polynomial ring R[x]. Integrality provides the answer to the question when the converse does hold. Note the analogy to field extensions, where finitely generated by algebraic elements equals finite-dimensional.

4.6 Proposition. Let R be a commutative ring and T a commutative R-algebra. Then the following are equivalent:

(1) T is finitely generated as an R-module.

(2) T is finitely generated over R a a ring and integral over R.

(3) T is finitely generated by integral elements over R as a ring.

Proof. We already know (1 ⇒ 2). (2 ⇒ 3) holds a fortiori. To see (3 ⇒ 1), assume T is generated as a ring over R by t_{1}, …, t_{m}. Then T is generated as an R-module by monomials t_{1}^{k_{1}}t_{2}^{k_{2}} … t_{m}^{k_{m}}. If each t_{i} is integral over R then for each i there exists an exponent n_{i} such that every power of t_{i} is expressible as an R-linear combination of 1, t, …, t^{n_{i}}, and therefore T is generated as an R-module by monomials t_{1}^{k_{1}}t_{2}^{k_{2}} … t_{m}^{k_{m}} with 0 ≤ k_{i} ≤ n_{i}. □

4.7 Corollary. Let R be a commutative ring, T an R-algebra and a, b ∈ T with ab = ba. If both R[a] and R[b] are finitely generated as R-modules then so is R[a, b]. In particular, if a and b are commuting integral elements over R, then a − b and ab are integral over R, too.

4.8 Corollary. Let R ⊆ T be commutative rings. Then the set of all elements of T integral over R form a ring.

Commutativity is essential here. If the integral elements a and b do not commute, neither their sum nor their product needs to be integral over R.

4.9 Definition. Let R ⊆ T be commutative rings. The ring

R′ = {t ∈ T | t is integral over R}

is called the integral closure of R in T. R is called integrally closed in T if R′ = R. “The” integral closure of a domain R is the integral closure in its quotient field. R is called integrally closed if R is integrally closed in its quotient field.

4.10 Proposition. Let R be a commutative ring, R ⊆ S an integral ring extension and u integral over S. Then u is integral over R.

Proof. Suppose u^{n} + a_{n−1}u^{n−1} + … + a_{0} = 0 with a_{0}, …, a_{n−1} ∈ S. Then R[a_{0}, …, a_{n−1}] is finitely generated integral over R. Let g_{1}, …, g_{m} be generators of R[a_{0}, …, a_{n−1}] as an R-module. Then R[a_{0}, …, a_{n−1}, u] is generated as an R-module by the elements g_{i}u^{k} for 1 ≤ i ≤ m and 1 ≤ k ≤ n − 1. □

4.11 Corollary (Transitivity of integral extensions). Let R ⊆ S ⊆ T be commutative rings. If T is integral over S and S is integral over T then T is integral over R.

4.12 Corollary (Integral closure is integrally closed.). Let R ⊆ T be commutative rings and R′ the integral closure of R in T. Then R′ is integrally closed in T.

4.13 Theorem. Let R be an integrally closed domain with quotient field K, L a field containing K, and u ∈ L algebraic over K. Then u is integral over R iff the minimal polynomial of u over K is in R[x].

Proof. Assume u integral over R. Let f be the minimal polynomial of u over K and L̄ its splitting field over L. Over L̄, f(x) = (x − u)(x − u_{1}) … (x − u_{m}). Since u and the u_{i} are roots of the same irreducible polynomial on K[x] there exists for each i an isomorphism φ: K[u] → K[u_{i}] with φ(u) = u_{i} and φ(k) = k for all k ∈ K. If u is a root of some monic g ∈ R[x], then 0 = φ(0) = φ(g(u)) = g(u_{i}) shows u_{i} to be a root of the same monic g ∈ R[x]. All u_{i} are integral over R. Therefore the coefficients of f (as sums of products of the roots) are also integral over R. At the same time, the coefficients are in K, therefore, R being integrally closed, the coefficients of f are in R. □

4.14 Corollary. Let R be a domain with quotient field K and u ∈ L algebraic over K. Then u is integral over R iff the coefficients of the minimal polynomial of u over K are integral over R.

4.15 Lemma. Let R ⊆ T be commutative rings and u ∈ T an invertible element. Then u^{−1} is integral over R if and only if u^{−1} ∈ R[u].

Proof. An equation showing u^{−1} to be integral over R, such as

(u^{−1})^{n} = a_{n−1}(u^{−1})^{n−1} + … + a_{1}(u^{−1}) + a_{0},

can be multiplied by u^{n−1} to show u^{−1} to be in R[u]:

u^{−1} = a_{n−1} + … + a_{1}u^{n−2} + a_{0}u^{n−1}.

Conversely, an equation showing u^{−1} ∈ R[u], can be multiplied by an appropriate power of u^{−1} to show u^{−1} integral over R. □

4.16 Proposition. Let R ⊆ L be an integral extension of domains. Then R is a field if and only if L is a field.

Proof. If R is a field then L is an algebraic extension of a field and therefore a field. If L is a field and u is a non-zero element of R then u has an inverse u^{−1} in L, which by the previous lemma is in R[u] = R. Therefore R is a field. □

4.17 Definition. Ler R ⊆ T be commutative rings. We introduce names for a few properties that the ring extension R ⊆ T may or may not satisfy:

Lying over. For every prime ideal P ⊴ R there exists a prime ideal P̃ ⊴ T with P̃ ∩ R = P. Going up. If P ⊆ Q are prime ideals of R and P̃ a prime ideal of T with P̃ ∩ R = P then there exists a prime ideal Q̃ of T with P̃ ⊆ Q̃ and Q̃ ∩ R = Q. Going down. If P ⊆ Q are prime ideals of R and Q̃ a prime ideal of T with Q̃ ∩ R = Q then there exists a prime ideal P̃ of T with P̃ ⊆ Q̃ and P̃ ∩ R = P. Incomparability. For any two prime ideals P̃ and Q̃ of T with P̃ ∩R = Q̃∩R neither P̃ ⊆ Q̃ nor Q̃ ⊆ P̃ holds.

Let R ⊆ T be commutative rings, P a prime ideal of R and S = R \ P. If Q is an ideal of T, then Q ∩ R ⊆ P if and only if Q ∩ S = ∅. Since S is a multiplicative subset of T containing (0), we know there exist ideals of T not intersecting S, and among them ideals maximal with this property, which are necessarily prime. The question is now whether an ideal maximal among those not intersecting S actually intersects R in P and vice versa. It turns out that the properties “incomparability” and “going up” can be characterized in this way.

4.18 Proposition. Let R ⊆ T be commutative rings. Then the following are equivalent:

(1) R ⊆ T satisfies “going up”.

(2) For every prime ideal P of R, if Q is an ideal of T maximal with respect to the property Q ∩ (R \ P) = ∅ then Q ∩ R = P.

Since ideals maximal with respect to avoiding a multiplicative set S with 0 ∉ S are guaranteed to exist and are prime, we get:

4.19 Corollary. “Going up” implies “lying over”.

4.20 Proposition. Let R ⊆ T be commutative rings. Then the following are equivalent:

(1) R ⊆ T satisfies “incomparability”.

(2) For every prime ideal P of R, if Q is a prime ideal of T with Q ∩ R = P then Q is maximal with respect to the property Q ∩ (R \ P) = ∅.

4.21 Theorem. Let R ⊆ T be an integral extension of rings. Then “incomparability” and “going up” (and therefore “lying over”) hold.

Proof. To show “going up”, we assume P is a prime ideal of R and P̃ an ideal of T maximal with respect to the property P̃ ∩(R\P) = ∅; we must show P̃ ∩R = P. Suppose otherwise and choose u ∈ P \ P̃. Then, by maximality of P̃, there exists an s ∈ (R \ P) ∩ (P̃ + T u), s = q + tu with q ∈ P̃ and t ∈ T. t is integral over R, i.e.,

t^{n} + c_{n−1}t^{n−1} + … + c_{0} = 0

with coeffiecients c_{i} ∈ R. Multiplying by u^{n}, we get

(tu)^{n} + c_{n−1}u(tu)^{n−1} + … + u^{n}c_{0} = 0.

Since tu + q = s, tu ≡ s mod P̃, and therefore

s^{n} + c_{n−1}us^{n−1} + … + u^{n}c_{0} ≡ 0 (mod P̃).

The left side of this congruence is in R, so it is actually in P̃ ∩ R ⊆ P. Since u ∈ P we get s^{n} ∈ P and therefore s ∈ P, a contradiction.

To show “incomparability”, we assume P̃ ∩ R = P and must show that P̃ is maximal with respect to not intersecting S = R \ P. Suppose there exists Q̃ with P̃ ⊂ Q̃ and Q̃ ∩ S = ∅. Choose u ∈ Q̃ \ P̃. u is integral over R. So in particular there exists a monic polynomial f in R[x] with f(u) ∈ P̃.

Let f(x) = x^{n} + ∑_{i=0}^{n−1} a_{i}x^{i} be of minimal degree with this property. Clearly, deg f = n ≥ 1. Since u^{n} + a_{n−1}u^{n−1} + … + a_{0} ∈ P̃ ⊆ Q̃ and u ∈ Q̃, it follows that a_{0} ∈ Q̃ ∩ R = P. Now u(u^{n−1} + a_{n−1}u^{n−2} + … + a_{1}) ∈ P̃, but no factor is in P̃, a contradiction. □

4.22 Remark. If R ⊆ T is an integral extension of domains and R is integrally closed then “going down” holds.

The length of a chain of ideals I_{0} ⊇ I_{1} ⊇ I_{2} ⊇ … is defined as the number of proper inclusions occurring in the chain.

4.23 Definition. Let P be a prime ideal in a ring R. The height of P is the maximal length of a chain of prime ideals descending from P: ht(P) = n ∈ ℕ_{0} if n is maximal such that there exists a chain of prime ideals P ⊃ P_{1} ⊃ … ⊃ P_{n}, and ht(P) = ∞ if there are chains of prime ideals of arbitrary length descending from P.)

4.24 Definition. The Krull dimension dim(R) of a ring R is the supremum of the heights of the prime ideals of R.

Note that a zero-dimensional commutative ring is either a field or else a ring with zero-divisors in which every prime ideal is maximal; and that an integral domain is one-dimensional if and only if every prime ideal other than (0) is maximal.

4.25 Theorem. Let R ⊆ T be commutative rings.

(1) If “incomparability” holds and P is a prime ideal of R then for every prime ideal Q of T with Q ∩ R = P we have ht(Q) ≤ ht(P).

(2) If “going up” holds and P is a prime ideal of R of finite height then there exists a prime ideal Q of R with Q ∩ R = P and ht(Q) ≥ ht(P).

4.26 Corollary. If both “incomparability” and “going up” hold for R ⊆ T (in particular, if R ⊆ T is an integral extension) then for every prime ideal P of R there exists a prime ideal Q of R with Q ∩ R = P and ht(Q) = ht(P).

Proof. Ad 1) Every chain of prime ideals descending from Q contracts to a chain of the same length descending from P, hence ht(P) ≥ ht(Q). Ad 2) “Lying over” and repeated applications of “going up” allow us to construct for every prime ideal chain descending from P a chain of prime ideals in T of the same length descending from some Q with Q ∩ R = P. □

4.27 Definition. The co-height of a prime ideal P of a commutative ring R is the maximal length of a chain of prime ideals ascending from P (in other words, co-ht(P) = dim(R/P).)

4.28 Theorem. Let R ⊆ T be commutative rings. If “going up” and “incomparability” hold (in particular, if R ⊆ T is an integral extension) then for every prime ideal P of R and prime ideal Q of T with Q∩R = P we have co-ht(Q) = co-ht(P).

Proof. Easy. □

4.29 Theorem. Let S be a multiplicative subset of a domain R. If R is integrally closed then so is S^{−1}R.

Proof. Let K be the quotient field of R and S^{−1}R, u ∈ K integral over S^{−1}R; to show u ∈ S^{−1}R.

u^{n} + \frac{a_{n−1}}{s_{n−1}} u^{n−1} + … + \frac{a_{0}}{s_{0}} = 0 a_{i} ∈ R, s_{i} ∈ S

By multiplication with s = s_{0} … s_{n−1} we get

su^{n} + r_{n−1}a_{n−1} + … + r_{0}a_{0} = 0 r_{i}, a_{i} ∈ R.

Multiplication with s^{n−1} shows su to be integral over R; therefore su ∈ R, and u ∈ S^{−1}R. □

4.30 Theorem. Let R_{α}, α ∈ A be a collection of integral domains all contained in a field K. If every R_{α} is integrally closed then so is ⋂_{α∈A} R_{α}.

Proof. Easy. □

Since every domain R satisfies R = ⋂_{P prime} R_{P} = ⋂_{Mmaximal} R_{M}, we see:

4.31 Corollary. Let R be an integral domain. Then the following are equivalent:

(1) R is integrally closed.

(2) For every prime ideal P of R, R_{P} is integrally closed.

(3) For every maximal ideal M of R, R_{M} is integrally closed.

A commutative ring R is called normal if for every prime ideal P of R the localization R_{P} is an integrally closed domain. We have seen that the concepts of normal and integrally closed coincide in the case of an integral domain.