In this paper, originally published in Linear and Multilinear Algebra 61 (9) (2013) 1161–1180, there was an error in the last step of the proof of Theorem 2.2. Let R be an algebra over a field 
, let U and V be left R-modules, and the dual spaces regarded as right modules over R and the space of all module homomorphisms from U to V. The suggestion that the general case can be reduced to finitely generated modules by inverse limits, does not work. The theorem is not true without assuming something about U or V. The following requires only a minor modification of arguments from the correct part of the proof.
Theorem 0.1
Let R be an -algebra (where is a field) and U, V any left R-modules. If U is finitely related (= a quotient of a free module by a finitely generated submodule) then each can be approximated by adjoints of elements of in the following sense: for every finite subsets G of U and H of there exists such thatThe same conclusion holds also if V is injective or if each finite subset of U is contained in a complemented finitely related submodule of U.
Since Theorem 2.2 has been used later in the paper, the results must be corrected as follows. Theorem 2.4 is wrong, from the above version of Theorem 2.2 we can deduce only the following corollary.
Corollary 0.2
If V is injective over R or if each finite subset of V is contained in a finitely related complemented submodule of V then
In the case when , the algebra of polynomials acting on V via for a fixed , the arguments from Section 3 describe the bicommutant of (that is, the center of ) and not necessarily (the center of ) and show that Z is contained in the set . The reverse inclusion is false in general, however, it is true if V as a module over satisfies the following condition:
V is torsion or V is injective or V contains a copy of R as a direct summand.
Thus, Theorem 2.4, Proposition 3.5, and Theorem 3.9 are valid for modules over
satisfying (A). The results in Section 4 (Theorems 4.1 and 4.3) are also valid under the hypothesis (A) or (more generally) if we assume that
.
A corrected version of the paper is posted on the arXiv.org.
