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I have the following statement:

For an arbitrary graded algebra $A$ generated by homogeneous elements of positive degree, the homogeneous elements $a_1,\dots,a_n$ form a minimal system of generators iff they form a basis of $\varepsilon(A)$ modulo $\varepsilon^2(A)$ (where $\varepsilon(A)$ denotes the augmentation ideal of $A$).

First, I know the description of augmentation ideal for a group ring (the kernel of augmentation map), then I think that the augmentation ideal for an associative algebra is the kernel of augmentation map of the associative algebra, but in this case I do not have the map explicity. Can I get this map explicitly? The augmentation map is any $K$-algebra homomorphism or has some conditions, forms...? If the augmentation ideal is the kernel of augmentation map why in this statement the ideal is denoted like the image of augmentation map?

What is the idea to show this statement? Do you know any book that treat it?

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  • $\begingroup$ Yes, $\varepsilon(A)$ is really bad notation. $\endgroup$ – darij grinberg Aug 18 '16 at 15:33
  • $\begingroup$ Actually he use $\omega$ instead of $\varepsilon$. But I think the meaning of $\omega$ is to denote the augmentation map. $\endgroup$ – donikvep Aug 18 '16 at 16:04

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