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?
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?