Abstract
Fuchs called a partially-ordered integral domain, say D, division closed if it has the property that whenever a > 0 and ab > 0, then b > 0. He showed that if D is a lattice-ordered division closed field, then D is totally ordered. In fact, it is known that for a lattice-ordered division ring, the following three conditions are equivalent: a) squares are positive, b) the order is total, and c) the ring is division closed. In the present article, our aim is to study \({\ell}\)-rings that possibly possess zerodivisors and focus on a natural generalization of the property of being division closed, what we call regular division closed. Our investigations lead us to the concept of a positive separating element in an \({\ell}\)-ring, which is related to the well-known concept of a positive d-element.
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Presented by C. Tsinakis.
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Ma, J., McGovern, W.W. Division closed partially ordered rings. Algebra Univers. 78, 515–532 (2017). https://doi.org/10.1007/s00012-017-0467-7
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DOI: https://doi.org/10.1007/s00012-017-0467-7