Ring homomorphism between quotients of rings of polynomials in many variables

Question

Let \begin{equation*}A=\mathbb{Z}[x_{ij}: 1 \leq i < j \leq n]/(x_{ij}x_{kl}=x_{ik}x_{jl}+x_{il}x_{kj} \; \text{for} \; 1\leq i < j < k <l \leq n) \end{equation*} and \begin{equation*} B=\mathbb{Z}[x_{ij}:1 \leq i < j \leq m]/(x_{ij}x_{kl}=x_{ik}x_{jl}+x_{il}x_{kj} \; \text{for} \; 1\leq i < j < k <l \leq m), \end{equation*} with $m < n$, where both $m,n \in \mathbb{N}$. I am trying to think of a ring homomorphism from $A$ to $B$ but I am not sure what should that look like. Intuitively, it should be some sort of projection map, but how would I define one rigorously?

Answer 1

We have maps $\pi_A: \mathbb{Z}[x_{ij}: 1 \leq i < j \leq n] \longrightarrow A$ and $\pi_B: \mathbb{Z}[x_{ij}: 1 \leq i < j \leq m] \longrightarrow B$.

Since $m<n$ we get a projection map (by forgetting some variables) $$\varphi: \mathbb{Z}[x_{ij}: 1 \leq i < j \leq n] \longrightarrow \mathbb{Z}[x_{ij}: 1 \leq i < j \leq m]$$

Now if we look at the composition $\pi_B \varphi:\mathbb{Z}[x_{ij}: 1 \leq i < j \leq n] \longrightarrow B$, we can see that $I:=(x_{ij}x_{kl}=x_{ik}x_{jl}+x_{il}x_{kj} \; \text{for} \; 1\leq i < j < k <l \leq n) \subset \ker(\pi_B\varphi)$.

This holds because $m<n$, so if we have the relations for $n$ we have them after projection for $m$.

Therefore we can use the universal property of quotient rings to get a unique morphism $\phi:A \longrightarrow B$, s.t. $\phi \pi_A =\pi_B \varphi$.

In particular this is a ring homomorphism, since all the other maps are ring homomorphisms as well.