Posted by Jason Polak on 21. September 2017 · Write a comment · Categories: modules · Tags:

Here is one characterisation of commutative rings of Krull dimension zero:

Theorem. A commutative ring $R$ has Krull dimension zero if and only if every element of the Jacobson radical ${\rm Jac}(R)$ of $R$ is nilpotent and the quotient ring $R/{\rm Jac}(R)$ is von Neumann regular.

Recall that a ring $R$ is von Neumann regular if for every $x\in R$ there exists a $y\in R$ such that $xyx = x$. This odd property is equivalent to saying that every $R$-module is flat.

Here are two examples of what happens when we drop various assumptions in the “if” direction of the theorem:

  1. The ring $\Z_{(p)}$ of integers localised away from the prime $(p)$ is an example of a ring such that $R/{\rm Jac}(R)$ is von Neumann regular but ${\rm Jac}(R)$ has no nontrivial nilpotent elements. The ring $\Z_{(p)}$ has Krull dimension one.
  2. Another type of example is given by $\Z[[t]]/t^n$ where $\Z[[t]]$ denotes the power series ring with integer coefficients. Unlike our first example, the Jacobson radical of this ring is the ideal $(t)$, which is also the nilradical (=set of nilpotent elements), but $R/{\rm Jac}(R) = \Z$, which is not von Neumann regular and has Krull dimension one.

Note that we were forced look for counterexamples to dropped assumptions in the class of infinite rings. That’s because every finite commutative ring has Krull dimension zero.

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