Analysis of A-Module Endomorphism \phi: Understanding Kronecker Delta Function

[sid_galt]

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

Then the book I have (commutative algebra by atiyah goes on to say) - That means
\sum_{j=1}^{n} (\delta_{ij}\phi - a_{ij})x_j=0,\ \delta_{ij} being the kronecker delta function This is the part I can't understand - how can you separate \phi form it's argument x_j. How can \phi(x) = \phi\cdot x?

 

[arnaldur]

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.