A trick question about rings

This semester I am lecturing the level-3 course Rings and Modules at QUB. While reviewing some material and preparing lecture notes I puzzled over something for a while before finally catching my mistake. How do you fare?

Question: Let $\mathbf{H}$ be the ring of quaternions — the four-dimensional real vector space with basis $1, i, j, k$ and the usual relations. Let $\mathbf{H}[X]$ be the ring of univariate polynomials with coefficients in $\mathbf{H}$ . What is $\mathbf{H}[X] / (X-i)$ ?

Take a moment to try to answer the question if you like before reading on.

The trap here is to think of quotienting by $(X-i)$ as meaning “substitute $X \mapsto i$ ”, which would suggest that the quotient ring is isomorphic to $\mathbf{H}$ . That’s the wrong answer. The problem is that the map defined by sending a polynomial $f(X) \in \mathbf{H}[X]$ to “ $f(i)$ ” is not a homomorphism! Indeed it is implicit in the definition of the ring of polynomials that $X$ commutes with the base ring, whereas the element $i$ does not.

In fact $\mathbf{H}[X] / (X-i)$ is the zero ring. To see this, let $I = (X-i)$ and observe that $I$ contains $-j(X - i) j = X + i$ and hence also $(X+i) - (X-i) = 2i$ , a unit, so $I = \mathbf{H}[X]$ .