Exercise 111.9.3 (076G)—The Stacks project

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Exercise 111.9.3. Compute the length of the following modules over the following rings. Briefly(!) explain your answer. (Please feel free to use additivity of the length function in short exact sequences, see Algebra, Lemma 10.52.3).

The length of $\mathbf{Z}/120\mathbf{Z}$ over $\mathbf{Z}$ .
The length of $\mathbf{C}[x]/(x^{100} + x + 1)$ over $\mathbf{C}[x]$ .
The length of $\mathbf{R}[x]/(x^4 + 2x^2 + 1)$ over $\mathbf{R}[x]$ .

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