typos on homework 1

In the first version of Problem 2(a), I had written F[x] \otimes F[y] \cong F[x,y]. The current formulation says F[x] \otimes F[x] \cong F[x,y]. These statements are equivalent (can you see why?), but I want you to prove the current formulation.

Problem 2(c) should say:

Prove that M_{m \times m}(F) \otimes M_{n \times n}(F) \cong M_{mn \times mn}(F)  as F-algebras.

Problem 3 should say: “A nonzero element x is grouplike if and only if x \in G.”

In Problem 5(c), the algebra is called $\wedge(V)$.





  1. Really Basic Undergrad Question, and I’ve been putting off puttering with it because there’s other stuff to be done: If f is injective, and f = g \circ h, then obviously f^{-1} = (g \circ h)^{-1}. However, can we conclude that both g and h are invertible?

    (This, btw, is related to something I was doing with the Universal Property and mappings)

  2. Sebastián O.

    In the first version of problem 2.a. we can state that \mathbb{F}[x] \otimes \mathbb{F}[x] \cong \mathbb{F}[x] \otimes \mathbb{F}[y] because \mathbb{F}[x] \cong \mathbb{F}[y] as \mathbb{F}-algebras, right?

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