Abstract Algebra – Show A1?R A2 is in Ring

abstract-algebracategory-theory

We denote $\textbf{Ring}$ the category of conmutative rings with $1$.

Let $R \in \textbf{Ring}$. Consider $\phi_1: R \to A_1$ and $\phi_2: R \to A_2$ ring morphisms.

I have to show three things:

  1. Show that the tensor product $A_1 \otimes_R A_2 \in \textbf{Ring}$ (considering $A_1$ and $A_2$ as $R-$modules).

  2. Show there are $\lambda_1: A_1 \to A_1 \otimes_R A_2$ and $\lambda_2: A_2 \to A_1 \otimes_R A_2$ ring morphisms.

  3. Show that the following square is a Pushout in $\textbf{Ring}$:

enter image description here

I'm a little bit confused with the writing of the exercise and with the idea of every required proof.

For the first proof, do we need to consider an arbitrary right $R-$module $A_1$ and an arbitrary left $R-$module $A_2$, in order to define the tensor product $A_1 \otimes_R A_2 $, and to show that $A_1 \otimes_R A_2 \in \textbf{Ring}$, right?

For the second proof, to find $\lambda_1$ and $\lambda_2$ morphisms in $\textbf{Ring}$, we need that $A_1, A_2 \in \textbf{Ring}$. But, for arbitrary $A_1$ and $A_2$ $R-$modules, why should $A_1, A_2 \in \textbf{Ring}$?

Best Answer

This is a good exercise, but is confusingly written. Here's a longer version of the same question, along with some hints, which should hopefully clarify the question and help you solve it!

For your first question: $A_1$ and $A_2$ are rings, and $\varphi_1 : R \to A_1$, $\varphi_2 : R \to A_2$ are ring homomorphisms. These maps give $A_1$ and $A_2$ the ~bonus structure~ of left/right $R$-modules, by using the multiplication in $A_1$ and $A_2$ to define

  • $r \cdot a_1 = \varphi_1(r) a_1$
  • $a_2 \cdot r = a_2 \varphi_2(r)$

Now we can take the tensor product in the category of $R$-modules, and get a new $R$-module $A_1 \otimes_R A_2$.

Then question (1) asks you to show that this object $A_1 \otimes_R A_2$, which we know is an $R$-module, is actually also a ring! I'll give you a hint that you should define the multiplication as $(a_1 \otimes a_2) (a_1' \otimes a_2') = a_1 a_1' \otimes a_2 a_2'$, using the multiplication in $A_1$ and $A_2$ in each slot. This also hopefully explains your confusin with question (2). You're right that it's not obvious that $A_1 \otimes_R A_2$ is a ring, but in question (1) you showed that it secretly is one!

Now that you've shown this, question (2) makes sense, and asks you to find ring morphisms $\lambda_1 : A_1 \to A_1 \otimes_R A_2$ and $\lambda_2 : A_2 \to A_1 \otimes_R A_2$. Again, I'll give you a hint that the right map to consider is $\lambda_1(a_1) = a_1 \otimes 1_{A_2}$ (and similarly for $\lambda_2$). Do you see how to show these are ring maps?

Then lastly, for question (3) you're meant to show that this square is a pushout in the category of rings. So you want to show that for any ring $S$ with ring maps $\alpha_1 : A_1 \to S$ and $\alpha_2 : A_2 \to S$ so that $\alpha_1 \circ \varphi_1 = \alpha_2 \circ \varphi_2 : R \to S$, there should be a unique ring map $A_1 \otimes_R A_2 \to S$. Again, I'll give you a hint that the correct map to consider is $a_1 \otimes a_2 \mapsto \alpha_1(a_1) \alpha_2(a_2)$, using the multiplication in $S$. Can you show this is well defined and a ring hom?

I hope this helps ^_^

Related Question