Past exam of the algebra, analysis and differential equations II course of the University of Oxford / 2009 / Algebra / 3 / a

The displayed matrix is block diagonal with a $1\times 1$ block $1$ and a $2\times 2$ block $B$ having characteristic polynomial $X^2-4X-5=(X-5)(X+1)$. The latter block is not scalar, so its minimal polynomial has degree two and is exactly that polynomial. Taking the least common multiple of the block minimal polynomials gives $m=(X-1)(X-5)(X+1)$ over $\mathbb{R}$.

Ideals of $K[X]/(m)$ correspond to ideals of $K[X]$ containing $(m)$, namely the principal ideals generated by the monic divisors of $m$. Every one of the three distinct linear factors may be included or omitted, soThis includes both the zero ideal and the whole ring. Equivalently the Chinese remainder theorem identifies this ring with ${\mathbb{R}}^3$.