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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
DOI: https://doi.org/10.1090/S0002-9947-1983-0684499-6
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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Keywords: Lattice-ordering ring, <IMG WIDTH="12" HEIGHT="19" ALIGN="BOTTOM" BORDER="0" SRC="images/img1.gif" ALT="$l$">-prime <IMG WIDTH="12" HEIGHT="19" ALIGN="BOTTOM" BORDER="0" SRC="images/img2.gif" ALT="$l$">-ring, nilpotent element, domain, squares positive, polynomial constraints
Article copyright: © Copyright 1983 American Mathematical Society