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

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.