Abstract
In this paper we prove that the Ziegler spectra of all serial rings are sober. We then use this proof to give a general framework for computing and understanding the T0-quotients of Ziegler spectra of uniserial rings. Finally, we illustrate this technique by computing the T0-quotients of Ziegler spectra of all rank one uniserial domains.
Highlights
The Ziegler spectrum, ZgR, of a ring R is a topological space attached to its module category of R
Soberness of Ziegler spectra was first studied by Herzog in [Her93] where he showed that every irreducible closed subset of ZgR with a countable neighbourhood basis of open sets is the closure of a point
Hochster, [Hoc69], showed that prime spectra of rings are exactly those topological spaces which are T0, quasi-compact, have a basis of compact open sets which is stable under intersection and which are sober
Summary
For general background on Model theory of Modules see [Pre88] or for a more algebraic perspective see [Pre09]. We write φ(M ) for the solution set of a pp-formula φ in an R-module M. We write pinjR for the set of isomorphism classes of indecomposable pure-injective modules. The Ziegler spectrum of a ring R, denoted ZgR, is a topological space whose points are isomorphism classes of indecomposable pure-injective modules and which has a basis of open sets given by:. [EH95, Corollary 1.6][Pun01a, Lemma 11.1] Let R be a serial ring and e1, ..., en a complete set of primitive orthogonal idempotents. [Pun01a, Lemma 11.2] Let R be a serial ring and e a primitive idempotent. [Pun01a, 11.8][EH95, Theorem 2.7] Let R be a serial ring, e ∈ R a primitive idempotent and I, J an e-pair. En, the above lemma implies that all indecomposable pure-injectives are of the form N (I, J) for some consistent ei-pair.
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