help in an example.

Question

In page 226 of David Eisenbud's book Commutative Algebra with a View Toward Algebraic Geometry there is an example which I need help in some parts of it:

why $codim I= 1$? why $dim M = dim R = 2$?

Answer 1

I'll use $x,y,z$ for the variables of $R$. Notice that $(x)(y,z) = (xy,xz)$. If we want to form a chain of prime ideals of $R$ terminating at $I$, then they must all contain $xy$ and $xz$. But this implies that they either contain $x$, or $y$ and $z$. If they contain $x$, then so too does $I$, but this is a contradiction. So they must contain $y,z$. But then clearly, $(y,z)\subseteq (x+1, y,z) = I$ is a maximal chain, so the height is $1$.

$M$ is the set of primes containing $x$. A maximal chain here is $(x)\subseteq(x,y)\subseteq(x,y,z)$ just as in the comments.