Ziegler spectrum
In mathematics, the (right) Ziegler spectrum of a ring "R" is a topological space whose points are (isomorphism classes of) indecomposable pure-injective right "R"-modules. Its closed subsets correspond to theories of modules closed under arbitrary products and direct summands. Ziegler spectra are named after Martin Ziegler, who first defined and studied them in 1984. Definition. 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 where formula_2 are natural numbers, formula_3 is an formula_4 matrix with entries from "R", and formula_5  is an formula_6-tuple of variables and formula_7 is an formula_8-tuple of variables. The (right) Ziegler spectrum, formula_9, of "R" is the topological space whose points are isomorphism classes of indecomposable pure-injective right modules, denoted by formula_10, and the topology has the sets as subbasis of open sets, where formula_12 range over (right) pp-1-formulae and formula_13 denotes the subgroup of formula_14 consisting of all elements that satisfy the one-variable formula formula_15. One can show that these sets form a basis. Properties. Ziegler spectra are rarely hausdorff and often fail to have the formula_16-property. However they  are always compact and have a basis of compact open sets given by the sets formula_17 where formula_12 are pp-1-formulae. When the ring "R" is countable formula_9 is sober. It is not currently known if all Ziegler spectra are sober. Generalization. Ivo Herzog showed in 1997 how to define the Ziegler spectrum of a locally coherent Grothendieck category, which generalizes the construction above.