Definition 2

An A-module M is called generating if every A-module N is generated by the images of the A-linear mappings from M to N.

Let M be a left A-module. We denote the dual of M by $M^*$ and the canonical bilinear form on $M\times M^*$ (II, §2, No. 3, p. 233) by

We denote by $\tau (M)$ the set of elements of A of the form $\sum^n_{i=1}\langle x_i, x^*_i\rangle$, where $x_1, . . . , x_n$ are elements of M and $x^*_1, . . . , x^*_n$ are elements of $M^*$. It is a two-sided ideal of A, called the trace ideal of M. The trace ideal of the A-module $A_s$ is A. The trace ideal of a module that is the direct sum of a family $(M_i)_{i\in I}$ of A-modules is the ideal $\sum_{i\in I}\tau (M_i)$. If M is a projective A-module, then it follows from Proposition 12 of II, §2, No. 6, p. 238, that we have $M =\tau (M)M$.