2006
DOI: 10.1016/j.jalgebra.2005.10.039
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On the existence of directed rings and algebras with negative squares

Abstract: We show that there exist many directed rings and algebras with negative squares.

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Cited by 12 publications
(13 citation statements)
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“…We show now that directed partial orders constructed in [2,3] are special cases of P x,y (S) constructed above.…”
Section: Theorem 26mentioning
confidence: 81%
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“…We show now that directed partial orders constructed in [2,3] are special cases of P x,y (S) constructed above.…”
Section: Theorem 26mentioning
confidence: 81%
“…In Section 2, we construct a class of directed partial orders on the complex numbers C via positive additive semigroups of F . This class of directed partial orders includes all directed partial orders on C constructed by Yang using negative valuations [3] and by Rump and Yang using multiplicative segment [2], and it contains the largest one among all directed partial orders on C. We conjecture that the directed partial orders that we have constructed give all possible directed partial orders on C such that 1 > 0. In Section 3, we use the similar construction to get directed partial orders on the quaternions over F .…”
Section: Introductionmentioning
confidence: 81%
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“…In [12], Schwartz showed that an Archimedean lattice-ordered (commutative) field that has 1 > 0 and that is algebraic over its maximal totally ordered subfield cannot have negative squares, and in [13], DeMarr and Steger showed that in a partially ordered finite dimensional real linear algebra no square can be the negative of a strong unit. Furthermore, in [15], we guarantee the existence of directed commutative fields with negative squares. In 1969, Conrad and Dauns [6] raised the following problem (this is Question (b) of their list in [6] [11], p. 124).…”
Section: Introductionmentioning
confidence: 99%