## The nonreduced order spectrum of a commutative ring.(English)Zbl 0941.14023

The authors introduce the order spectrum as a new spectral space associated with a commutative ring. It contains the real spectrum as a subspace. The points of the order spectrum of a ring $$A$$ are pairs $$\gamma= (I_\gamma, \leq_\gamma)$$ where $$I_\gamma$$ is an ideal of $$A$$ and $$\leq_\gamma$$ is a total order of the residue ring $$A/I_\gamma$$. The real spectrum consists of those points for which $$I_\gamma$$ is even a prime ideal. The paper motivates the introduction of the order spectrum. Some of its basic algebraic and topological properties, as well as connections with the real spectrum are explored. The general theory is illustrated by a number of elementary examples.

### MSC:

 14P10 Semialgebraic sets and related spaces 13J25 Ordered rings 06F25 Ordered rings, algebras, modules

### Keywords:

total order; order spectrum; real spectrum
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