Let $R$ be a Noetherian commutative ring and let $\dim(R)$ denote the Krull dimension of $R$. For the polynomial ring $R[x]$, we have $\dim(R[x]) = 1 + \dim(R)$. In fact, the same is true if we replace the polynomial ring by the power series ring: again $\dim(R[[x]]) = 1 + \dim(R)$. The situation is a […]
There are all sorts of notions of dimension that can be applied to rings. Whatever notion you use though, the ones with dimension zero are usually fairly simple compared with the rings of higher dimension. Here we’ll look at three types of dimension and state what the rings of zero dimension look like with respect […]
Let $R$ be a ring. The projective dimension $\mathrm{pd}_R(M)$ of an $R$-module $M$ is the infimum over the lengths of projective resolutions of $M$. The left global dimension of $R$ is the supremum over the projective dimensions of all left $R$-modules. There is a notion of right global dimension where left modules are replaced with […]