c](V) from the free

associative algebra of V to the free associative coalgebra of V, such that [[OHM].

and Iruing R, Representation Theory of Finite Groups and

Associative Algebra, John Wiley and Sons (1962).

One checks that M(A) is itself an

associative algebra with unit element under the following operations:

By a topological algebra we mean a topological vector space A over K, which is also an

associative algebra over K such that the multiplication in A is separately continuous (1).

The universal enveloping algebra U(ST(2)) of the Lie algebra ST(2) is the

associative algebra,

Consequently, Rota-Baxter systems yield pre-Lie and

associative algebra structures.

xn) be a free

associative algebra with n generators x1xn over a field K.

Hochschild, On the cohomology groups of an

associative algebra, Ann.

L]) with an

associative algebra u(R, L), an algebra map [[iota].

An identity of an

associative algebra is a non-commuting polynomial that vanishes identically on all substitutions in the algebra, say Kanel-Belov, Karasik, and Rowen, and an identity is a polynomial identity (PI) if at least one of its coefficients is more or less than one.

q]) on the

associative algebra of strictly upper triangular matrices.

We have a classical example of such an algebra when we consider the Hochschild cohomology of an

associative algebra [1].