I am working through some problems which require me to show when some $k$-algebra ($k$ a field) maps are isomorphisms. Unfortunately, I've got myself a bit confused with definitions and the like, and I don't quite know how to show when two $k$-algebras are isomorphic in practice.

Here's my definition: Let $A$ and $B$ be $k$-algebras with respective structure maps $a: k \rightarrow A$, $b: k \rightarrow B$. A $k$-algebra isomorphism is a ring isomorphism $f: A \rightarrow B$ s.t. $f \circ a = b$.

So, does this mean that, in practice, if I'm given a well-defined map $f$ from between two $k$-algebras, all I have to show is that it's a ring isomorphism? Or is there a more subtle point that requires checking?

Some examples of the maps I'm trying to show are $k$-algebra isomorphisms:

1) $f : M_n(k) \otimes_k A \rightarrow M_n(A)$, where $(x_{ij}) \otimes a \mapsto (ax_{ij})$. (Here, $A$ is any $k$-algebra).

2) $g: A \otimes_k k[X] \rightarrow A[X]$, where $a \otimes \sum r_i T^i \mapsto \sum r_i a T^i $. (Again, $A$ is any $k$-algebra)

If someone could explain how to go about explicitly showing when a map is a $k$-algebra isomorphism / demonstrate with one of those examples, it would be greatly appreciated!

  • $\begingroup$ 1 need not be an isomorphism and for 2 there are issues with commutativity. $\endgroup$ – Matt Samuel Apr 26 '15 at 13:31
  • $\begingroup$ I have replaced 1 with the closest thing that is correct. OP, is this what you meant? $\endgroup$ – darij grinberg Apr 26 '15 at 13:32
  • $\begingroup$ 2 is correct if you define $A[X]$ appropriately. $\endgroup$ – darij grinberg Apr 26 '15 at 13:33
  • $\begingroup$ Thank you for the edit @darijgrinberg , that was a typo on behalf. $\endgroup$ – user143137 Apr 26 '15 at 13:51
  • $\begingroup$ I have been told that these maps are definitely $k$-algebra isomorphisms, and the task is to verify that they are explicitly. $\endgroup$ – user143137 Apr 26 '15 at 13:52

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