By a careful investigation of the model theory of modules over a special class of uniserial domains we give some (counter) examples to a decomposition of a serial module. For instance there is a uniserial module M over a uniserial domain that is not quasi-small. Also there is a projective non–free countably generated module over the endomorphism ring of M . MSC: 16D70; 16D99; 03C60

In this paper we study almost uniserial rings and modules. An R−module M is called almost uniserial if any two nonisomorphic submodules are linearly ordered by inclusion. A ring R is an almost left uniserial ring if R_R is almost uniserial. We give some necessary and sufficient condition for an Artinian ring to be almost left uniserial.

The existence of valuation domains admitting non-standard uniserial modules for which certain Exts do not vanish was proved in [1] under Jensen’s Diamond Principle. In this note, the same is verified using the ZFC axioms alone. In the construction of large indecomposable divisible modules over certain valuation domainsR, the first author used the property thatR satisfied Ext1R(Q,U) 6= 0, where ...

All Lie algebras and representations will be assumed to be finite dimensional over the complex numbers. Let V (m) be the irreducible sl(2)module with highest weight m ≥ 1 and consider the perfect Lie algebra g = sl(2) n V (m). Recall that a g-module is uniserial when its submodules form a chain. In this paper we classify all uniserial g-modules. The main family of uniserial g-modules is actuall...

 ‎In this article‎, ‎we first‎ ‎show that non-Noetherian Artinian uniserial modules over‎ ‎commutative rings‎, ‎duo rings‎, ‎finite $R$-algebras and right‎ ‎Noetherian rings are $1$-atomic exactly like $Bbb Z_{p^{infty}}$‎. ‎Consequently‎, ‎we show that if $R$ is a right duo (or‎, ‎a right‎ ‎Noetherian) ring‎, ‎then the Noetherian dimension of an Artinian‎ ‎module with homogeneous uniserial dim...

let r be a ring, m a right r-module and (s,≤) a strictly ordered monoid. in this paper we will show that if (s,≤) is a strictly ordered monoid satisfying the condition that 0 ≤ s for all s ∈ s, then the module [[ms,≤]] of generalized power series is a uniserial right [[rs,≤]] ]]-module if and only if m is a simple right r-module and s is a chain monoid.

Let R be a ring, M a right R-module and (S,≤) a strictly ordered monoid. In this paper we will show that if (S,≤) is a strictly ordered monoid satisfying the condition that 0 ≤ s for all s ∈ S, then the module [[MS,≤]] of generalized power series is a uniserial right [[RS,≤]] ]]-module if and only if M is a simple right R-module and S is a chain monoid.

If R̂ is the pure-injective hull of a valuation ring R, it is proved that R̂ ⊗R M is the pure-injective hull of M , for every finitely generated Rmodule M . Moreover R̂ ⊗R M ∼= ⊕1≤k≤nR̂/AkR̂, where (Ak)1≤k≤n is the annihilator sequence of M . The pure-injective hulls of uniserial or polyserial modules are also investigated. Any two pure-composition series of a countably generated polyserial module a...