Does there exist a totally ordered ring with zero divisors? I can't think of an example right now.
2026-04-08 15:45:29.1775663129
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Existence of totally ordered ring with zero divisors
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Assuming the definition used in Fuchs, Partially Ordered Algebraic Systems, take the set of integer numbers with the usual order relation and the usual addition, and define the multiplication to be trivial, so that every product be $0$. Rings with trivial multiplication are one of the examples of ordered rings given in Fuchs's book.
L. Fuchs, Partially Ordered Algebraic Systems. Pergamon Press, 1963
Chapt VIII, $\S 3$ "O-rings with divisors of zero".