Module question

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Let [tex]A, M[/tex] be a commutative ring and a finitely generated A-module respectively. Let [tex]\phi[/tex] be an A-module endomorphism of M such that [tex]\phi (M)\subseteq \alpha\ M[/tex] where [tex]\alpha[/tex] is an ideal of A. Let [tex]x_1,\dots,x_n[/tex] be the generators of M. Then we know that [tex]\displaystyle{\phi(x_i)=\sum_{j=1}^{n} a_{ij}x_j\ (1\leq i\leq n;\ a_{ij}\in \alpha)}[/tex].

Then the book I have (commutative algebra by atiyah goes on to say) - That means
[tex] \sum_{j=1}^{n} (\delta_{ij}\phi - a_{ij})x_j=0,\ \delta_{ij}[/tex] being the kronecker delta function This is the part I can't understand - how can you separate [tex]\phi[/tex] form it's argument [tex]x_j[/tex]. How can [tex] \phi(x) = \phi\cdot x[/tex]?
 
The equation you quote from the book, is correct but the term within parenthesis is a mapping which is applied to xj instead of multiplying it. The aij part is actually aij times the identity map.
 
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