MATH 612 Due Wednesday April 14, 1993

1. Let R and R′ be rings.

a) Prove that there exists a multiplication on R ⊗_{ℤ} R′ such that (r_{1} ⊗ r′_{1})(r_{2} ⊗ r′_{2}) = r_{1}r_{2} ⊗ r′_{1}r′_{2} . Thus R ⊗_{ℤ} R′ can be considered as a ring.

b) Prove that if R and R′ are commutative, then the diagram

ℤ ⟶ R′
↓  ↓
R ⟶ R ⊗_{ℤ} R′

is a push-out in the category of commutative rings. (Here the maps out of ℤ are the only ones possible and the bottom horizontal map, for instance, is given by r ↦ r ⊗ 1.)

2. Let A be an R-algebra and let f ∈ R[X] and B = R[X]/(f). By abuse of notation, we will also use f to denote its image in A[X]. Prove that A ⊗_{R} B ≈ A[X]/(f).

3. If ℂ is the field of complex numbers, prove that i ⊗ 1 + 1 ⊗ i is a zero divisor in ℂ ⊗_{ℝ} ℂ.

4. Let E and F be extensions of a field k, let m = [E : k] and n = [F : k]. (These may possibly be infinite cardinals.) Then we can think of E ⊗_{k} F as either a k -algebra, an E -algebra, or an F -algebra.

a) Prove that dim_{E} E ⊗_{k} F = n, dim_{F} E ⊗_{k} F = m and dim_{k} E ⊗_{k} F = mn.

b) Prove that if e_{1}, …, e_{r} are elements of E linearly independant over k, and f_{1}, …, f_{s} are k -linearly independant elements of F, then the rs elements e_{1} ⊗ f_{1} , e_{1} ⊗ f_{2}, …, e_{r} ⊗ f_{s} of E ⊗_{k} F are linearly independant over k.

c) Prove that if F ⊆ F′ are fields then the inclusion F ↪ F′ induces a monomorphism E ⊗_{k} F → E ⊗_{k} F′, so that it is reasonable to think of E ⊗_{k} F as a k -subalgebra of E ⊗_{k} F′.

d) If E and F are both contained in an extension field K of k, prove that E and F are linearly disjoint over k [Lang, §X.5, p. 379] [Hungerford, §VI.2, p. 318] ⟺ E ⊗_{k} F is a field, and in this case E ⊗_{k} F ≈ EF.