Lattices of normally indecomposable modules

Brzezinski, Juliusz

doi:10.1090/S0002-9939-1978-0469979-4

Lattices of normally indecomposable modules
HTML articles powered by AMS MathViewer

by Juliusz Brzezinski PDF

Proc. Amer. Math. Soc. 68 (1978), 271-276 Request permission

Abstract:

If M, N are finitely generated left R-modules, then M divides ${N^1}$ if there is an epimorphism ${M^{(r)}} \to N$. M is normally indecomposable if $M \cong {M_1} \oplus {M_2}$ and ${M_1}$ divides ${M_2}$ imply ${M_2} = 0$. If R is an Artin algebra or an order over a complete discrete valuation ring in a semisimple algebra, the set of isomorphism classes of normally indecomposable R-modules (respectively R-lattices) is partially ordered by the divisibility relation. We show that for R of finite representation type this partially ordered set is a lattice satisfying the Jordan-Dedekind chain condition and the length of maximal chains is equal to the number of isomorphism classes of indecomposable R-modules (respectively R-lattices).

References

Garrett Birkhoff, Lattice theory, 3rd ed., American Mathematical Society Colloquium Publications, Vol. XXV, American Mathematical Society, Providence, R.I., 1967. MR 0227053
Peter Gabriel, Unzerlegbare Darstellungen. I, Manuscripta Math. 6 (1972), 71–103; correction, ibid. 6 (1972), 309 (German, with English summary). MR 332887, DOI 10.1007/BF01298413
Peter Gabriel, Indecomposable representations. II, Symposia Mathematica, Vol. XI (Convegno di Algebra Commutativa, INDAM, Rome, 1971) Academic Press, London, 1973, pp. 81–104. MR 0340377
A. Heller and I. Reiner, Indecomposable representations, Illinois J. Math. 5 (1961), 314–323. MR 122890
A. V. Roĭter, Divisibility in the category of representations over a complete local Dedekind ring, Ukrain. Mat. Ž. 17 (1965), no. 4, 124–129 (Russian). MR 0197534