## Positive derivations on Archimedean lattice-ordered rings.(English)Zbl 1167.06012

A lattice-ordered ring $$A$$ is called an $$f$$-ring if $$x\land y=0$$ and $$0\leq z$$ imply $$zx\land y=xz\land y=0$$ for all $$x,y,z\in A$$. It is well-known that the only positive derivation (i.e., one that maps positive elements to positive elements) on a reduced Archimedean $$f$$-ring is the zero derivation, cf. [P. Colville, G. Davis and K. Keimel, J. Aust. Math. Soc., Ser. A 23, 371–375 (1977; Zbl 0376.06021)].
The paper under review examines the situation for general Archimedean lattice-ordered rings. The beginning sections focus on derivatives and derivations on certain polynomial and group rings. For instance, it is shown that for group rings of finite cyclic groups the only derivation vanishing on the underlying ring is the zero derivation. For infinite cyclic groups such derivations are always based on the derivative, see Corollary 3.4 or Theorem 3.2 for a more general statement.
In the second half of the article the authors turn their attention to derivatives and derivations on Archimedean lattice-ordered rings. It is proved that on purely transcendental extensions of totally ordered fields derivations that are at the same time lattice homomorphisms, are translations of the usual derivative (Theorem 9.1), and on algebraic extensions of totally ordered fields often the only positive derivation is the zero derivation (see Section 6 for detailed statements). The authors also give some results for lattice-ordered matrix rings (Section 8) and lattice-ordered rings in which all squares are positive (Section 7).
The paper abounds with examples that illustrate the results and show the necessity of hypotheses.

### MSC:

 06F25 Ordered rings, algebras, modules 13J25 Ordered rings 16W25 Derivations, actions of Lie algebras 46G05 Derivatives of functions in infinite-dimensional spaces

Zbl 0376.06021
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### References:

 [1] M. Anderson, T. Feil Lattice-Ordered Groups, Reidel, Dordrecht, 1988, ISBN 90-277-2643-4. [2] A. Bigard, K. Keimel, S. Wolfenstein, Groupes et Anneuax Réticulés, LNM 608, Springer-Verlag, Berlin, 1977, ISBN 3-540-08436-3. [3] G. Birkhoff, Lattice Theory, Colloq. Pub., 25, 3rd edn., Am. Math. Soc., Providence, (1973). [4] G. Birkhoff, R. S. Pierce, Lattice-ordered rings, A. Acad. Bras., 28 (1956), 41–69. · Zbl 0070.26602 [5] F. F. Bonsall, J. Duncan, Complete Normed Algebras, Springer-Verlag, New York, 1973, ISBN 0-387-06386-2. · Zbl 0271.46039 [6] K. Boulabiar, Positive derivations on archimedean almost f-rings, Order, 19 (2002), 385–395. · Zbl 1025.06015 [7] M. C. Bourlet, Sur les opérations en général et les équations différentielles linéaires d’ordre infini, Ann. Ec. Norm. Sup., 14(3) (1897), 133–190. · JFM 28.0350.02 [8] P. Colville, G. Davis, K. Keimel, Positive derivations on f-rings, J. Aust. Math. Soc. 23 (1977), 371–375. · Zbl 0376.06021 [9] T.-Y. Dai, Positive derivations and homomorphisms on partially ordered linear algebra, Algebra and Order, In: S. Wolfenstein (ed.) Proceedings of the International Conference on Ordered Algebraic Structures, Luminy-Marseilles, 1984, Hermann Verlag, Berlin (1986), 203–215. [10] T.-Y. Dai, R. DeMarr, Positive derivations on partially ordered linear algebra with an Order Unit, Proc. Am. Math. Soc. 72 (1978), 21–26. · Zbl 0402.46006 [11] M. R. Darnel, Theory of Lattice-Ordered Groups, Marcel Dekker, Inc., New York, 1995, ISBN 0-8247-9326-9. · Zbl 0810.06016 [12] J. E. Diem, A radical for lattice-ordered rings, Pac. J. Math., 25 (1968), 71–82. · Zbl 0157.08004 [13] L. Fuchs, Partially Ordered Algebraic Systems, Pergamon Press, Oxford, (1963). · Zbl 0137.02001 [14] A. M. W. Glass, Partially Ordered Groups, World Scientific, Singapore, 1999, ISBN 981-02-3493-3497. · Zbl 0933.06010 [15] D. J. Hansen, Positive derivations on partially ordered strongly regular rings, J. Aust. Math. Soc., (Series A) 37 (1984), 178–180. · Zbl 0558.06021 [16] M. Henriksen, F. A. Smith, Some properties of positive derivations on f-rings, Contemp. Math., 8 (1982), 175–184. · Zbl 0491.06016 [17] I. M. Isaacs, Algebra, a graduate course, Brooks/Cole Publishing Company, Pacific Grove, 1994, ISBN 0-534-19002-2. · Zbl 0805.00001 [18] N. Jacobson, Basic Algebra I, 2nd edn., Freeman, New York (1985). · Zbl 0557.16001 [19] J. B. Miller, Higher derivations on Banach algebras, Am. J. Math., 92 (1970), 301–331. · Zbl 0201.17203 [20] J. Ma, P. Wojciechowski, Lattice orders on matrix algebras, Algebra Univers., 47 (2002), 435–441. · Zbl 1059.06015 [21] R. H. Redfield, Unexpected lattice-ordered quotient structures, In: W. C. Holland (ed.) Ordered Algebraic Structures: Nanjing (2001), Gordon & Breach, Amsterdam, 2001, ISBN 90-5699-325-9, 111–131. · Zbl 1038.06513 [22] S. A. Steinberg, Unital -prime lattice-ordered rings with polynomial constraints are domains, Trans. Am. Math. Soc., 276 (1983), 145–164. · Zbl 0506.06007 [23] R. R. Wilson, Lattice orderings on the real field, Pac. J. Math. 63 (1976), 571–577. · Zbl 0297.12101
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