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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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  • [1] A. Bigard, Contribution à la théorie des groupes réticulé, Thèse Sci. Math., Paris, 1969. MR 0250950 (40:4181)
  • [2] A. Bigard, K. Keimel and S. Wolfenstein, Groupes et anneaux réticulé, Lecture Notes in Math., vol. 608, Springer-Verlag, Berlin and New York, 1977. MR 0552653 (58:27688)
  • [3] G. Birkhoff, Lattice theory, 3rd ed., Amer. Math. Soc. Colloq. Publ., vol. 25, Amer. Math. Soc, Providence, R.I., 1968.
  • [4] G. Birkhoff and R. S. Pierce, Lattice-ordered rings, An. Acad. Brasil. Ci. 28 (1956), 41-69. MR 0080099 (18:191d)
  • [5] J. E. Diem. A radical for lattice-ordered rings, Pacific J. Math. 25 (1968), 71-82. MR 0227068 (37:2653)
  • [6] L. Fuchs, Partially ordered algebraic systems, Pergamon Press, New York, 1963. MR 0171864 (30:2090)
  • [7] M. Henriksen and J. Isbell, Lattice-ordered rings and function rings, Pacific J. Math. 12 (1962), 533-565. MR 0153709 (27:3670)
  • [8] I. N. Herstein, Noncommutative rings, Carus Math. Monographs 15 (1968). MR 0227205 (37:2790)
  • [9] D. G. Johnson, A structure theory for a class of lattice-ordered rings, Acta Math. 104 (1960), 163-215. MR 0125141 (23:A2447)
  • [10] R. S. Pierce, Radicals in function rings, Duke Math J. 23 (1956), 253-261. MR 0078958 (18:6b)
  • [11] M. A. Shatalova, $ {l_A}$ and $ {l_I}$ rings, Siberian Math. J. 7 (1966), 1084-1094.
  • [12] H. J. Shyr and T. M. Viswanathan, On the radicals of lattice-ordered rings, Pacific J. Math. 54 (1974), 257-260. MR 0376482 (51:12657)
  • [13] S. A. Steinberg, Finitely-valued $ f$-modules, Pacific J. Math. 40 (1972), 723-737. MR 0306078 (46:5205)
  • [14] -, Identities and nilpotent elements in lattice ordered rings, Ring Theory (S. K. Jain, ed.), Dekker, New York, 1977. MR 0491403 (58:10659)
  • [15] -, On lattice-ordered rings in which the square of every element is positive, J. Austral. Math. Soc. 22 (1976), 362-370. MR 0427198 (55:233)
  • [16] -, Examples of lattice-ordered rings, J. Algebra 72 (1981), 223-236. MR 634624 (83b:06018)

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Additional Information

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

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