－1－
Minimal Prime Ideals of a Tensor Product of Algebras over a Field Mem. Fac. Educ., Kagawa Univ. II, 64（2014）, １－ ４
Minimal Prime Ideals of a Tensor Product of Algebras
over a Field
Kazunori F
UJITAINTRODUCTION. All rings considered in this paper are assumed to be commutative with identity, and all ring homomorphisms are unital. We use Spec（A） and Min（A） to denote the sets of prime ideals, and minimal prime ideals, respectively, of a ring A.
Throughout this note, k will denote a field. Let K be an extension field of k, then Ks
denotes the maximal separable algebraic extension of k. The field extension K/k is said to be primary if every element of K which is separable algebraic over k is contained in
k, i.e. if Ks = k. Let K and L be two extensions of a field k. Then the map P ⟼ Pc is a
homeomorphism between Min（K⊗k L） and Spec（Ks⊗k Ls） （［3, Theorem 3］）.
If the minimal prime ideals in a ring A are pairwise comaximal, we say that A
satisfies MPC （for Minimal Primes Comaximality）. A ring is called quasi normal if it is integrally closed in its total quotient ring. In ［1, Proposition 4.2］ it is proved that if a quasi-normal ring A has no imbedded prime divisor of zero, then A is the direct sum of a finite number of quasi-normal noetherian rings Ai such that each Ai is either an integrally
closed domain or an Artin local ring.
LEMMA 1. ［3, Proposition 2］. Let R be a k-algebra, K/k a primary field extension and
T = K⊗k R. Then the canonical map: P ⟼ P ∩ R, P ∈ Spec（T）, is a bijection between
Min（T） and Min（R）.
THEOREM 2. Let K1, K2, · · ·, Kn be extension fields of k. Then the map P ⟼ Pc is a
homeomorphism between Min（K1⊗k · · · ⊗k Kn） and Spec（（K1）s⊗k · · · ⊗（Kk n）s）.
PROOF. Let A0 = （K1）s ⊗k · · · ⊗k （Kn）s, Ai := K1⊗k · · · ⊗k Ki ⊗（Kk i+1）s ⊗k · · · ⊗k （Kn）s
（i = 1, · · ·, n）. Ki / （Ki）s is a primary field extension and Ki⊗（Ki）s Ai−1 ≅ Ai. Therefore the
canonical map Min（Ai） ⟶ Min（Ai−1） is a bijection by Lemma 1 （i = 1, · · ·, n）. Since
dim（A0） = 0, Min（A0） = Spec（A0）. Hence, the map Min（An） ⟶ Spec（A0） is a bijection.
Now Min（An） is compact, and Spec（A0） is Hausdorf, so the map Min（An） ⟶ Spec（A0）
－2－ Kazunori FUJITA
PROPOSITION 3. Let K1, K2, · · ·, Kn be extension fields of k. Then K1⊗k · · · ⊗k Knsatisfies MPC.
PROOF. This follows almost immediately from Theorem 2.
LEMMA 4. Let k be an algebraically closed field. Let A and B be k-algebras. Let μ : Spec （A ⊗k B） ⟶ Spec（A）×Spec（B） be the map with μ（P） = （P ∩ A, P ∩ B）. Let p and q
be prime ideals of A and B respectively, and j＊: Spec（k（p）⊗k k（q）） ⟶ Spec（A⊗
k B） be the map induced by the canonical k-algebra homomorphism j : A⊗k B ⟶ k（p）⊗k （q）, k where k（p） = Ap/pAp, k（q） = Aq/qAq. Then,
（1） p（A⊗k B） + q（A⊗k B） is a prime ideal of A⊗k B.
（2） Im（j＊） = μ−1（（p, q））
（3） p（A⊗k B） + q（A⊗k B） is the smallest element of μ−1（（p, q））.
PROOF. （1） （A⊗k B） / （p（A⊗k B） + q（A⊗k B）） ≅ （A/p）⊗（B/q） is an integral domain k
by ［5, Corollary 1, p.198］, so p（A⊗k B） + q（A⊗k B） is a prime ideal of A⊗k B.
（2） and （3） follow from ［4, Proposition 2.3］.
PROPOSITION 5 （［2, Theorem 2.3］）. Let k be an algebraically closed field. Let A and B be
k-algebras. If A and B satisfy MPC, then A⊗k B satisfies MPC.
PROOF. This is immediate from Lemma 4. PROPOSITION 6. Let A, B be k-algebras.
（1） If A is an integrally closed domain and B is a zero dimensional local ring, then
A⊗k B satisfies MPC.
（2） If A and B are zero dimensional local rings, then A⊗k B satisfies MPC.
PROOF. （1） Let N be the maximal ideal of B. Then N = nil（B）, so the canonical map Spec（A⊗（B/N）） k ⟶ Spec（A⊗k B） is a bijection. It follows from ［2, Theorem 3.4］ that A⊗（B/N） satisfies MPC. Therefore A⊗k k B satisfies MPC.
（2） Let M and N be the maximal ideals of A and B respectively. Then the canonical map Spec（（A/M）⊗（B/N）） k ⟶ Spec（A⊗k B） is a bijection. It follows from Proposition 2 that
（A/M）⊗（B/N） satisfies MPC. Therefore A⊗k k B satisfies MPC.
A ring is called quasi normal if it is integrally closed in its total quotient ring. THEOREM 7. Let A and B be quasi-normal noetherian rings that are k-algebras. Suppose
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Minimal Prime Ideals of a Tensor Product of Algebras over a Field
PROOF. By ［1, Proposition 4.2］, we may assume that A = A1×· · ·×Am, B = B1×· · ·×Bn
where each Ai（resp. Bj） is either an integrally closed domain or an Artin local ring. So A⊗k B = Πi, j（Ai⊗k Bj） satisfies MPC by ［2, Theorem 3.4］ and Proposition 6.
EXAMPLE There exists a field extension L of k and a noeterian domain A over k such that
L ⊗k A does not satisfy MPC.
Let L be a finite Galois extension field of a field k with L = k（α）, ［L : k］ = n > 1. Let B := L［X］（X）, P := XB and A := k + P. Let m（Y） be the minimal polynomial for α
over k, and α1 = α, α2, · · ·, αn be the roots of the equation m（Y） = 0. Let ϕi : A［Y］ ⟶ B
be the A-algebra homomorphisms such that ϕ（Y） = αi i, and let ψi : B［Y］ ⟶ B be the B-algebra homomorphisms such that ψ（Y） = αi （i = 1, · · ·, n）. Let pi i := Ker（ϕi）, Pi :=
Ker（ψi）. Then,
（1） B = L + P
（2） B = A + Aα + Aα2 + · · · + Aαn−1, and hence A is noetherian by Eakin-Nagataʼs
theorem.
（3） B = S −1（L［X］） where S = k［X］－Xk［X］.
（4） L ⊗k A≅ A［Y］ / （m（Y））.
（5） Pi = （Y － αi） B ［Y］, pi = Pi∩A［Y］ （i = 1, · · ·, n）.
（6） M = m（Y）A［Y］ + PA［Y］ is a maximal ideal of A［Y］.
（7） Ni = （Y － αi）B［Y］ + PB［Y］ is a maximal ideal of B［Y］, and Ni ∩ A［Y］ = M
（i = 1, · · ·, n）.
（8） p1 /（m（Y））, · · ·, pn /（m（Y）） are minimal prime ideals of A［Y］/（m（Y））, and p1 /（m（Y）） + · · · + pn /（m（Y）） M /（m（Y））.
（9） L ⊗k A does not satisfy MPC.
References
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［2］ S. Bouchiba, D. E. Dobbs and S-E. Kabbaj, On the prime ideal structure of tensor products of algebras, J. Pure Apple. Algebra 176 （2002）, 89－112.
［3］ P. Vamos, On the minimal prime ideals of a tensor product of two fields. London. Math. Proc. Camb. Phil. Soc. 84 （1978）, 25－35.
［4］ Adrian R. Wadsworth, The Krull dimensions of tensor products of commutative algebras over a field, J. London Math. Soc. 19 （1979）, 391－401.
－4－ Kazunori FUJITA
［5］ O. Zariski and P. Samuel, Commutative Algebra, Vol. I, Van Nostrand, Princeton, 1958.
