Fall 2016 Problem 6. Show that the ℝ-modules L = ℂ ⊗ℝ ℂ and L = ℂ ⊗ℂ ℂ are not isomorphic.

Fall 2013 Problem 2. Consider an attempt to make an ℝ-linear map

f : ℂ ⊗ℂ ℂ → ℂ ⊗ℝ ℂ

or

ℂ ⊗ℝ ℂ → ℂ ⊗ℂ ℂ,

in either direction given be the formula f(x ⊗ y) = x ⊗ y. In which direction is this map well defiined? Is it then surjective? Is it injective?

Fall 2018 Problem 2. Let ℍ be the real quarternions. Then ℂ ⊗ℝ ℍ is isomorphic to which of the following rings? Prove your answer.

(a) ℂ × ℂ

(b) ℂ × ℂ × ℂ × ℂ

(c) M_{2}(ℂ)

(d) M_{2}(ℝ)

(e) M_{2}(ℍ)

(f) M_{2}(ℝ) × M_{2}(ℝ)

Fall 2014 Problem 6. If R is a commutative ring with identity, and S a multiplicative subset, then every ideal J of S^{−1}R is of the form S^{−1}I for some ideal I of R. Is I uniquely determined by J? Why or why not?

Winter 2008 Problem 5: What are the possible orders of the tensor product as ℤ modules of two Abelian groups which each have order 25?

Winter 2009 Problem 4: Let P be any set of primes in ℤ. Find a commutative ring R containing ℤ for which the irreducible elements are precisely units times elements of P. (Hint: consider ℤ[\frac{1}{2}] = \{\frac{a}{2^{r}} \} ⊆ ℚ).

Fall 2009 Problem 5: Show that ℂ ⊗ℝ ℂ and ℂ ⊕ ℂ are isomorphic as ℂalgebras.

Spring 2014 Problem 6: Consider the field ℂ(x) of complex rational functions as a ℂ-algebra and the tensor product of two copies ℂ(x) ⊗ℂ ℂ(y) as another ℂalgebra. Is this also a field?

Fall 2018 Problem 4. Solve the following:

(a) Prove that R/I⊗_{R}R/J ≅ R/(I+J) for R a commutative ring, and I, J ⊂ R are ideals.

(b) Find the dimension of ℚ[x, y]/(x^{2} + y^{2}) ⊗_{ℚ[x,y]} ℚ[x, y]/(x + y^{3}) as a vector space over ℚ, or explain why it is infinite.