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

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- by Stuart A. Steinberg PDF
- Trans. Amer. Math. Soc.
**276**(1983), 145-164 Request permission

## 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.## References

- Alain Bigard,
*Groupes archimĂ©diens et hyper-archimĂ©diens*, SĂ©minaire P. Dubreil, M.-L. Dubreil-Jacotin, L. Lesieur et G. Pisot: 1967/68, AlgĂ¨bre et ThĂ©orie des Nombres, SecrĂ©tariat mathĂ©matique, Paris, 1969, pp.Â Fasc. 1, Exp. 2, 13 (French). MR**0250950** - Alain Bigard, Klaus Keimel, and Samuel Wolfenstein,
*Groupes et anneaux rĂ©ticulĂ©s*, Lecture Notes in Mathematics, Vol. 608, Springer-Verlag, Berlin-New York, 1977 (French). MR**0552653**, DOI 10.1007/BFb0067004
G. Birkhoff, - Garrett Birkhoff and R. S. Pierce,
*Lattice-ordered rings*, An. Acad. Brasil. Ci.**28**(1956), 41â€“69. MR**80099** - J. E. Diem,
*A radical for lattice-ordered rings*, Pacific J. Math.**25**(1968), 71â€“82. MR**227068**, DOI 10.2140/pjm.1968.25.71 - L. Fuchs,
*Partially ordered algebraic systems*, Pergamon Press, Oxford-London-New York-Paris; Addison-Wesley Publishing Co., Inc., Reading, Mass.-Palo Alto, Calif.-London, 1963. MR**0171864** - Melvin Henriksen and J. R. Isbell,
*Lattice-ordered rings and function rings*, Pacific J. Math.**12**(1962), 533â€“565. MR**153709**, DOI 10.2140/pjm.1962.12.533 - I. N. Herstein,
*Noncommutative rings*, The Carus Mathematical Monographs, No. 15, Mathematical Association of America; distributed by John Wiley & Sons, Inc., New York, 1968. MR**0227205** - D. G. Johnson,
*A structure theory for a class of lattice-ordered rings*, Acta Math.**104**(1960), 163â€“215. MR**125141**, DOI 10.1007/BF02546389 - R. S. Pierce,
*Radicals in function rings*, Duke Math. J.**23**(1956), 253â€“261. MR**78958**, DOI 10.1215/S0012-7094-56-02323-7
M. A. Shatalova, ${l_A}$ - H. J. Shyr and T. M. Viswanathan,
*On the radicals of lattice-ordered rings*, Pacific J. Math.**54**(1974), no.Â 1, 257â€“260. MR**376482**, DOI 10.2140/pjm.1974.54.257 - Stuart A. Steinberg,
*Finitely-valued $f$-modules*, Pacific J. Math.**40**(1972), 723â€“737. MR**306078**, DOI 10.2140/pjm.1972.40.723 - Stuart A. Steinberg,
*Identities and nilpotent elements in lattice-ordered rings*, Ring theory (Proc. Conf., Ohio Univ., Athens, Ohio, 1976) Lecture Notes in Pure and Appl. Math., Vol. 25, Dekker, New York, 1977, pp.Â 191â€“212. MR**0491403** - Stuart A. Steinberg,
*On lattice-ordered rings in which the square of every element is positive*, J. Austral. Math. Soc. Ser. A**22**(1976), no.Â 3, 362â€“370. MR**427198**, DOI 10.1017/s1446788700014804 - Stuart A. Steinberg,
*Examples of lattice-ordered rings*, J. Algebra**72**(1981), no.Â 1, 223â€“236. MR**634624**, DOI 10.1016/0021-8693(81)90319-7

*Lattice theory*, 3rd ed., Amer. Math. Soc. Colloq. Publ., vol. 25, Amer. Math. Soc, Providence, R.I., 1968.

*and*${l_I}$

*rings*, Siberian Math. J.

**7**(1966), 1084-1094.

## Additional Information

- © Copyright 1983 American Mathematical Society
- 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