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( n , d ) -injective covers, n -coherent rings, and ( n , d ) -rings

Weiqing Li, Baiyu Ouyang (2014)

Czechoslovak Mathematical Journal

It is known that a ring R is left Noetherian if and only if every left R -module has an injective (pre)cover. We show that ( 1 ) if R is a right n -coherent ring, then every right R -module has an ( n , d ) -injective (pre)cover; ( 2 ) if R is a ring such that every ( n , 0 ) -injective right R -module is n -pure extending, and if every right R -module has an ( n , 0 ) -injective cover, then R is right n -coherent. As applications of these results, we give some characterizations of ( n , d ) -rings, von Neumann regular rings and semisimple rings....

A duality result for almost split sequences

Lidia Hügel, Helmut Valenta (1999)

Colloquium Mathematicae

Over an artinian hereditary ring R, we discuss how the existence of almost split sequences starting at the indecomposable non-injective preprojective right R-modules is related to the existence of almost split sequences ending at the indecomposable non-projective preinjective left R-modules. This answers a question raised by Simson in [27] in connection with pure semisimple rings.

A note on test modules

Ladislav Bican, Pavel Jambor, Tomáš Kepka, Petr Němec (1976)

Commentationes Mathematicae Universitatis Carolinae

Almost split sequences for non-regular modules

S. Liu (1993)

Fundamenta Mathematicae

Let A be an Artin algebra and let 0 → X → ⊕ i = 1 r Y i → Z → 0 be an almost split sequence of A-modules with the Y i indecomposable. Suppose that X has a projective predecessor and Z has an injective successor in the Auslander-Reiten quiver Γ A of A. Then r ≤ 4, and r = 4 implies that one of the Y i is projective-injective. Moreover, if X → ⊕ j = 1 t Y j is a source map with the Y j indecomposable and X on an oriented cycle in Γ A , then t ≤ 4 and at most three of the Y j are not projective. The dual statement for a sink map holds. Finally, if an arrow...

Almost-flat modules

Simion Breaz (2003)

Czechoslovak Mathematical Journal

We present general properties for almost-flat modules and we prove that a self-small right module is almost flat as a left module over its endomorphism ring if and only if the class of g -static modules is closed under the kernels.

Automorphism liftable modules

Chelliah Selvaraj, Sudalaimuthu Santhakumar (2018)

Commentationes Mathematicae Universitatis Carolinae

We introduce the notion of an automorphism liftable module and give a characterization to it. We prove that category equivalence preserves automorphism liftable. Furthermore, we characterize semisimple rings, perfect rings, hereditary rings and quasi-Frobenius rings by properties of automorphism liftable modules. Also, we study automorphism liftable modules with summand sum property (SSP) and summand intersection property (SIP).

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