# Derivations and the First Cohomology Group of Trivial Extension Algebras

Derivations and the First Cohomology Group of Trivial Extension Algebras In this paper, we investigate, in detail, derivations on trivial extension algebras. We obtain generalizations of both known results on derivations on triangular matrix algebras and a known result on first cohomology group of trivial extension algebras. As a consequence, we get the characterization of trivial extension algebras on which every derivation is inner. We show that, under some conditions, a trivial extension algebra on which every derivation is inner has necessarily a triangular matrix representation. The paper starts with detailed study (with examples) of the relation between the trivial extension algebras and the triangular matrix algebras. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mediterranean Journal of Mathematics Springer Journals

# Derivations and the First Cohomology Group of Trivial Extension Algebras

, Volume 14 (4) – Jun 13, 2017
18 pages

/lp/springer_journal/derivations-and-the-first-cohomology-group-of-trivial-extension-2Wnwj05ZJW
Publisher
Springer International Publishing
Subject
Mathematics; Mathematics, general
ISSN
1660-5446
eISSN
1660-5454
D.O.I.
10.1007/s00009-017-0949-z
Publisher site
See Article on Publisher Site

### Abstract

In this paper, we investigate, in detail, derivations on trivial extension algebras. We obtain generalizations of both known results on derivations on triangular matrix algebras and a known result on first cohomology group of trivial extension algebras. As a consequence, we get the characterization of trivial extension algebras on which every derivation is inner. We show that, under some conditions, a trivial extension algebra on which every derivation is inner has necessarily a triangular matrix representation. The paper starts with detailed study (with examples) of the relation between the trivial extension algebras and the triangular matrix algebras.

### Journal

Mediterranean Journal of MathematicsSpringer Journals

Published: Jun 13, 2017

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