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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Unital $ l$-prime lattice-ordered rings with polynomial constraints are domains


Author: Stuart A. Steinberg
Journal: Trans. Amer. Math. Soc. 276 (1983), 145-164
MSC: Primary 16A86; Secondary 06A12, 06F25
MathSciNet review: 684499
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Abstract: It is shown that a unital lattice-ordered ring in which the square of every element is positive must be a domain provided the product of two nonzero $ l$-ideals is nonzero. More generally, the same conclusion follows if the condition $ {a^2} \geqslant 0$ is replaced by $ p(a) \geqslant 0$ for suitable polynomials $ p(x)$; and if it is replaced by $ f(a,b) \geqslant 0$ for suitable polynomials $ f(x,y)$ one gets an $ l$-domain. It is also shown that if $ a \wedge b = 0$ in a unital lattice-ordered algebra which satisfies these constraints, then the $ l$-ideals generated by $ ab$ and $ ba$ are identical.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1983-0684499-6
PII: S 0002-9947(1983)0684499-6
Keywords: Lattice-ordering ring, $ l$-prime $ l$-ring, nilpotent element, domain, squares positive, polynomial constraints
Article copyright: © Copyright 1983 American Mathematical Society