Socle deformations of self-injective algebras. (English) Zbl 0862.16001
Two Artin algebras \(\Lambda\) and \(R\) over a commutative Artin ring \(k\) are called right socle equivalent, if the factor algebras \(\Lambda/\text{soc}(\Lambda_\Lambda)\) and \(R/\text{soc}(R_R)\) are isomorphic. The paper deals with the question when a self-injective algebra \(A\) is socle equivalent to a split-extension algebra of an algebra \(B\), without oriented cycles in its ordinary quiver, by a \(B\)-bimodule. The authors introduce the notion of a deforming ideal of a self-injective algebra, investigate general properties of such ideals and exhibit their importance for the representation theory of self-injective Artin algebras.
Reviewer: W.Müller (Bayreuth)
MSC:
16D50 | Injective modules, self-injective associative rings |
16G70 | Auslander-Reiten sequences (almost split sequences) and Auslander-Reiten quivers |
16S70 | Extensions of associative rings by ideals |
16P20 | Artinian rings and modules (associative rings and algebras) |
16G10 | Representations of associative Artinian rings |