Mediterr. J. Math.
Springer International Publishing AG 2017
Derivations and the First Cohomology
Group of Trivial Extension Algebras
Driss Bennis and Brahim Fahid
Abstract. In this paper, we investigate, in detail, derivations on triv-
ial 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 deriva-
tion is inner. We show that, under some conditions, a trivial extension
algebra on which every derivation is inner has necessarily a triangu-
lar matrix representation. The paper starts with detailed study (with
examples) of the relation between the trivial extension algebras and the
triangular matrix algebras.
Mathematics Subject Classiﬁcation. 16W25, 15A78, 16E40.
Keywords. trivial extension algebra, triangular matrix algebra,
derivation, inner derivation, module generalized derivation, central inner
bimodule homomorphism, cohomology group.
Throughout the paper, R will denote a commutative ring with unity, A will
be a unital R-algebra with center Z(A), and M will be a unital A-bimodule.
Recall that an R-linear map D from A into M is said to be a derivation
if D(ab)=D(a)b + aD(b) for all a, b ∈A. It is known that the sum of
two derivations in A with values in M is also a derivation. This deﬁnes
the structure of a group on the set of all derivations in A with values in M
denoted by Der(A, M). In particular, when M = A, we simply set Der(A):=
Der(A, A). A derivation D∈Der(A, M) is said to be inner if it is of the form
,a] for some m
∈M, where [−, −] stands for the Lie bracket.
In addition, it is known that the set of all inner derivations in A with values
in M is a subgroup of Der(A, M). It will be denoted by Innder(A, M), and
when M = A, we simply set Innder(A) := Innder(A, A). It is a well-known
fact that a derivation needs not to be inner. Namely, the well-known ﬁrst