Length of a module, intersection theory

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I have a question to Proposition 7.1 "Intersection Theory", Fulton:

What is the length of $A/J$?

The length $l_R(M)$ is defined for an $R$-module $M$. Over which ring do I view $A/J$?

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$A/J$ can be viewed as a module over $A$ or $A/J$; the length is the same.