Suppose is a ring that has the invariant basis number property, and assume is a module that has a finite free resolution (that is, a free resolution of finite length by finitely generated free modules). Define the Euler characteristic of to be the Euler characteristic of any given finite free resolution:

has characteristic .

- Show the above is well defined, that is, it doesn’t depend on the given free resolution.
- Show that the Euler characteristic of any module over a noetherian or over a commutative ring is always non-negative.
- Exhibit a module over some ring that has negative Euler characteristic.

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