Failure of the Krull-Schmidt property for artinian modules and for serial modules

Alberto Facchini

Dipartimento di Matematica e Informatica, Università di Udine 33100 Udine, Italy

We shall consider unital modules over associative rings with identity. A module is uniserial if its lattice of submodules is linearly ordered under inclusion. A module is serial if it is a direct sum of uniserial submodules.

Both the class of artinian modules and the class of serial modules of finite Goldie dimension are classes of modules with semilocal endomorphism rings. If a module M_{R} has a semilocal endomorphism ring End(M_{R}), then all direct sum decompositions of M_{R} have only finitely many nonzero direct summands, and there are only finitely many direct sum decompositions of M_{R} up to isomorphism. For classes for which the Krull-Schmidt property holds, e.g. the class of all modules M_{R} that are finite direct sums of modules with local endomorphism rings, there is only one direct sum decomposition of M_{R} up to isomorphism.

It is possible to prove that every module-finite algebra over a semilocal noetherian commutative ring may be realized as the endomorphism ring of a suitable artinian module over a suitable ring. It follows that the Krull-Schmidt property fails for the class of artinian modules.

For what concerns serial modules the situation is the following. If M and N are two modules, we say that M and N belong to the same monogeny class, and write [M]_{m} = [N]_{m}, if there exist a monomorphism M → N and a monomorphism N → M. Similarly, we say that M and N belong to the same epigeny class, and write [M]_{e} = [N]_{e}, if there exist an epimorphism M → N and an epimorphism N → M.

Theorem. [Weak Krull-Schmidt Theorem for serial modules of finite Goldie dimension] Let U_{1}, …, U_{n}, V_{1}, …, V_{t} be uniserial modules over an arbitrary ring. Then the direct sums U_{1} ⊕…⊕U_{n} and V_{1} ⊕…⊕V_{t} are isomorphic if and only if n = t and there are two permutations σ, τ of {1, 2, …, n} such that [U_{σ(i)}]_{m} = [V_{i}]_{m} and [U_{τ(i)}]_{e} = [V_{i}]_{e} for every i = 1, 2, …, n.

This allows to show that the Krull-Schmidt property fails for the class of serial modules of finite Goldie dimension.

References

[1] N. V. Dung and A. Facchini, Weak Krull-Schmidt for infinite direct sums of uniserial modules, J. Algebra 193 (1997), 102–121.

[2] N. V. Dung and A. Facchini, Direct summands of serial modules, J. Pure Appl. Algebra, 1998, to appear.

[3] A. Facchini, Krull-Schmidt fails for serial modules, Trans. Amer. Math. Soc. 348 (1996), 4561–4575.

[4] A. Facchini, “Module Theory. Endomorphism rings and direct sum decompositions in some classes of modules”, Progress in Mathematics, Birkhäuser Verlag, Basel, 1998, to appear.

[5] A. Facchini and D. Herbera, K_{0} of a semilocal ring, preprint, 1998.

[6] A. Facchini, D. Herbera, L. S. Levy and P. Vámos, Krull-Schmidt fails for artinian modules, Proc. Amer. Math. Soc. 123 (1995), 3587–3592.

[7] W. Krull, Matrizen, Moduln und verallgemeinerte Abelsche Gruppen im Bereich der ganzen algebraischen Zahlen, Heidelberger Akademie der Wissenschaften 2 (1932), 13–38.

[8] R. B. Warfield, Jr., Serial rings and finitely presented modules, J. Algebra 37 (1975), 187–222.