Abstract
Let C be a finite dimensional algebra with B a split extension by a nilpotent bimodule E. We provide a short proof to a conjecture by Assem and Zacharia concerning properties of \(\mathop {\text {mod}}B\) inherited by \(\mathop {\text {mod}}C\). We show if B is a tilted algebra, then C is a tilted algebra.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Assem, I., Marmaridis, N.: Tilting modules over split-by-nilpotent extensions. Commun. Algebra 26, 1547–1555 (1998)
Assem, I., Simson, D., Skowronski, A.: Elements of the Representation Theory of Associative Algebras, 1: Techniques of Representation Theory, London Mathematical Society Student Texts, vol. 65. Cambridge University Press, Cambridge (2006)
Assem, I., Zacharia, D.: On split-by-nilpotent extensions. Colloq. Math. 98, 259–275 (2003)
Jaworska, A., Malicki, P., Skowroński, A.: Tilted algebras and short chains of modules. Math. Z. 273, 19–27 (2013)
Liu, S.: Tilted algebras and generalized standard Auslander-Reiten components. Arch. Math. 61, 12–19 (1993)
Skowroński, A.: Generalized standard Auslander-Reiten components without oriented cycles. Osaka J. Math. 30, 515–527 (1993)
Author information
Authors and Affiliations
Corresponding author
Additional information
The author was supported by the University of Connecticut-Waterbury.
Rights and permissions
About this article
Cite this article
Zito, S. Short proof of a conjecture concerning split-by-nilpotent extensions. Arch. Math. 111, 479–483 (2018). https://doi.org/10.1007/s00013-018-1208-7
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00013-018-1208-7