D4-Modules

Abstract

A module [Formula: see text] is called a [Formula: see text]-module if, whenever [Formula: see text] and [Formula: see text] are submodules of [Formula: see text] with [Formula: see text] and [Formula: see text] is a homomorphism with [Formula: see text], we have [Formula: see text]. The class of [Formula: see text]-modules contains the [Formula: see text]-modules as well as the dual-square-free (DSF) modules. Furthermore, a [Formula: see text]-module [Formula: see text] is called pseudo-discrete if [Formula: see text] is also a lifting module. In this paper, we study the [Formula: see text]-, the DSF, and the pseudo-discrete modules, and show that a pseudo-discrete module is clean iff it has the finite exchange property iff it has the full exchange property.

D3-MODULES VERSUSD4-MODULES – APPLICATIONS TO QUIVERS

“…Proof. (i) ⇔ (ii) This follows from [7,Proposition 2.23] and the fact that injective modules have the finite internal exchange property.…”

“…For the equivalences of (i)-(vi) see [16,Proposition 5.6]. (vii) ⇒ (ii) follows by [7,Proposition 2.11(2)]. (iv) ⇒ (vii) Note that M ⊕ M is a nonsingular divisible module since M is nonsingular and divisible.…”

“…Conversely, to show that M is a D3-module (resp. D4-module), let A and B be two direct summands of M such that M = A + B (and [7,Theorem 2.2]). REMARK 2.5.…”

“…It was shown in [7,Proposition 2.23] that for a module M with the finite internal exchange property, M is a D3-module if and only if M is a D4-module. REMARK 2.12.…”

On a question concerning D4-modules

“…(For a detailed background of these notions, we refer to [6,Chapter 4] and to [7].) A module M is also called a lifting module if it satisfies condition D1 (see [8] for detailed information regarding these modules).…”