Let R be a ring (associative, with 1, not necessarily commutative). A (right) pp-n-formula is a formula in the language of (right) R-modules of the form

$${\displaystyle \exists {\overline {y}}\ ({\overline {y}},{\overline {x}})A=0}$$

 

where ${\displaystyle \ell ,n,m}$  are natural numbers, ${\displaystyle A}$  is an ${\displaystyle (\ell +n)\times m}$  matrix with entries from R, and ${\displaystyle {\overline {y}}}$  is an ${\displaystyle \ell }$ -tuple of variables and ${\displaystyle {\overline {x}}}$  is an ${\displaystyle n}$ -tuple of variables.

The (right) Ziegler spectrum, ${\displaystyle \operatorname {Zg} _{R}}$ , of R is the topological space whose points are isomorphism classes of indecomposable pure-injective right modules, denoted by ${\displaystyle \operatorname {pinj} _{R}}$ , and the topology has the sets

$${\displaystyle (\varphi /\psi )=\{N\in \operatorname {pinj} _{R}\mid \varphi (N)\supsetneq \psi (N)\cap \varphi (N)\}}$$

 

as subbasis of open sets, where ${\displaystyle \varphi ,\psi }$  range over (right) pp-1-formulae and ${\displaystyle \varphi (N)}$  denotes the subgroup of ${\displaystyle N}$  consisting of all elements that satisfy the one-variable formula ${\displaystyle \varphi }$ . One can show that these sets form a basis.

Ziegler spectra are rarely Hausdorff and often fail to have the ${\displaystyle T_{0}}$ -property. However they are always compact and have a basis of compact open sets given by the sets ${\displaystyle (\varphi /\psi )}$  where ${\displaystyle \varphi ,\psi }$  are pp-1-formulae.

When the ring R is countable ${\displaystyle \operatorname {Zg} _{R}}$  is sober.^[2] It is not currently known if all Ziegler spectra are sober.

Ivo Herzog showed in 1997 how to define the Ziegler spectrum of a locally coherent Grothendieck category, which generalizes the construction above.^[3]