If a polynomial identity guarantees that every partial order on a ring can be extended, then this identity is true only for a zero-ring

@article{Kreinovich1995IfAP,
  title={If a polynomial identity guarantees that every partial order on a ring can be extended, then this identity is true only for a zero-ring},
  author={Vladik Kreinovich},
  journal={algebra universalis},
  year={1995},
  volume={33},
  pages={237-242}
}
In his classical 1963 book on partially ordered algebraic systems, L. Fuchs formulated the following problem (No. 29). It is known that if an abelian groupG (i.e., a group that satisfies an identityxy=yx) can be linearly ordered, then every partial order onG can be extended to a linear order. Fuchs asked whether there exists a similar polynomial identityP=0 for (associative) rings. In other words, does there exist a polynomialP with a following property: if a ringR satisfies the identityP=0… Expand
A characterization of rings in which each partial order is contained in a total order
Rings in which each partial order can be extended to a total order are called O∗rings by Fuchs. We characterize O∗rings as subrings of algebras over the rationals that arise by freely adjoining anExpand
On lattice extensions of partial orders of rings
We propose the study of rings, every partial ordering of which extends to a lattice ordering (i∗-rings). We show techniques enabling us, in some important cases, to decide whether a ring is or is notExpand
F*-Rings Are O*
TLDR
The condition on the ordering of the ring is weakened by requiring that every partial order on R extends to an f-order, and the two classes of rings coincide. Expand

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