An example of an indecomposable module without non-zero hollow factor modules

We construct an indecomposable module M without non-zero hollow factor modules, showing that there are hollow-lifting modules which are not lifting. The existences of such modules had been left open in a recent work by N. Orhan, D. Keskin-Tutuncu and R. Tribak. | Turk J Math 31 (2007) , 415 – 419. ¨ ITAK ˙ c TUB An Example of an Indecomposable Module Without Non-Zero Hollow Factor Modules Christian Lomp Abstract A module M is called hollow-lifting if every submodule N of M such that M/N is hollow contains a direct summand D ⊆ N such that N/D is a small submodule of M/D. A module M is called lifting if such a direct summand D exists for every submodule N . We construct an indecomposable module M without non-zero hollow factor modules, showing that there are hollow-lifting modules which are not lifting. The existences of such modules had been left open in a recent work by N. Orhan, D. Keskin-T¨ ut¨ unc¨ u and R. Tribak [2]. Key Words: Hollow modules, Indecomposable modules, Lifting modules, coalgebras. 1. Introduction The purpose of this note is to give an example of an indecomposable module without non-zero hollow factor modules. The existences of such modules had been left open in a recent work by N. Orhan, D. Keskin-T¨ ut¨ unc¨ u and R. Tribak. Recall that a module M is called hollow if there are no two proper submodules K, L of M whose sum spans the whole module, . M = K + L. In [2], the authors where concerned with so-called hollow-lifting modules, . modules M that have the lifting property with respect to submodules N of M with M/N being hollow. Any module who does not admit such submodules N would be of course an example of a hollow-lifting module. AMS Mathematics Subject Classification: Primary 16D90, 16D70 415 LOMP 2. The Example Fix a field k and a finite set Σ with at least two elements. Denote by Σ∗ the set of all words in Σ and denote the empty word by ω. Let R be the set of functions f : Σ∗ → k. For any word w ∈ Σ∗ denote by πw the function with πw (u) = 0 if u = w and πw (u) = 1 if u = w. R becomes an associative ring with unit by pointwise addition and the convolution product f(u)g(v) f ∗ g(w) = uv=w for f, g ∈ R and w ∈ Σ∗ . The unit of R is πω . Let M be the k-vector space with basis Σ∗ ;