FINITE LATTICES OF PRERADICALS AND FINITE REPRESENTATION TYPE RINGS

Abstract

In this paper we study some classes of rings which have a finite lattice of preradicals. We characterize commutative rings with this condition as finite representation type rings, i.e., artinian principal ideal rings. In general, it is easy to see that the lattice of preradicals of a left pure semisimple ring is a set, but it may be infinite. In fact, for a finite dimensional path algebra Λ over an algebraically closed field we prove that Λ-pr is finite if and only if its quiver is a disjoint union of finite quivers of type An; hence there are path algebras of finite representation type such that its lattice of preradicals is an infinite set. As an example, we describe the lattice of preradicals over Λ = kQ when Q is of type An and it has the canonical orientation

Keywords

• : Preradical,finite representation type

References

1. [1] I. Assem, D. Simson and A. Skowronski, Elements of the Representation Theory
2. of Associative Algebras, Vol.1, London Mathematical Society Student
3. Texts, 65, Cambridge University Press, Cambridge, 2006.
4. [2] M. Auslander, Representation theory of Artin algebras II, Comm. Algebra, 1
5. (1974), 269–310.
6. [3] M. Auslander, I. Reiten and S. O. Smalø, Representation Theory of Artin Algebras,
7. Cambridge Studies in Advanced Mathematics, 36, Cambridge University Press, Cambridge, 1997.
8. [4] L. Bican, T. Kepka and P. Nemec, Rings, Modules and Preradicals, Lecture

9. Notes in Pure and Applied Mathematics, 75, Marcel Dekker, Inc., New York,1982.
10. [5] G. Birkhoff, Lattice Theory, Colloquium Publications XXV, American Mathematical
11. Society, New York, 1948.
12. [
13. [6] D. Eisenbud and P. Griffith, The structure of serial rings, Pacific J. Math., 36(1971), 109–121.
14. [7] R. Fern´andez-Alonso and S. Gavito, The lattice of preradicals over local uniserial
15. rings, J. Algebra Appl., 5(6) (2006), 731–746.
16. [8] P. Gabriel, Unzerlegbare Darstellungen I, Manuscripta Math., 6 (1972), 71–103.
17. [9] G. Gr¨atzer, General Lattice Theory, Birkh¨auser Verlag, Basel, 2003.
18. [10] L. Gruson and C. U. Jensen, Deux applications de la notion de L-dimension,
19. C. R. Acad. Sci. Paris, S´er. A-B, 282(1) (1976), 23–24.
20. [11] B. Huisgen-Zimmermann, Purity, algebraic compactness, direct sum decompositions
21. and representation type, Infinite length modules, Trends Math.,(2000), 331–367.
22. [12] F. Raggi, J. R. Montes, H. Rinc´on, R. Fern´andez-Alonso and C. Signoret, The
23. lattice structure of preradicals, Comm. Algebra, 30(3) (2002), 1533–1544.
24. [13] F. Raggi, J. R´ıos, H. Rinc´on, R. Fern´andez-Alonso and C. Signoret, The lattice
25. structure of preradicals II: Partitions, J. Algebra Appl., 1(2) (2002), 201–214.
26. [14] R. P. Stanley, Enumerative Combinatorics, Vol. I, Wadsworth and Brooks/Cole
27. Advanced Books and Software, Monterey, CA, 1986.
28. [15] B. Stenstr¨om, Rings of Quotients, Die Grundlehrem der Mathematischen Wissenschaften
29. 217, Springer-Verlag, New York-Heidelberg, 1975.
30. [16] B. Zimmermann-Huisgen and W. Zimmermann, On the sparsity of representations
31. of rings of pure global dimension zero, Trans. Amer. Math. Soc., 320(2)
32. (1990), 695–711.