Definition 111.9.1. Let $A$ be a ring. Let $M$ be an $A$-module. The length of $M$ as an $R$-module is In other words, the supremum of the lengths of chains of submodules.
\[ \text{length}_ A(M) = \sup \{ n \mid \exists \ 0 = M_0 \subset M_1 \subset \ldots \subset M_ n = M, \text{ }M_ i \not= M_{i + 1} \} . \]
Exercise 111.9.2. Show that a module $M$ over a ring $A$ has length $1$ if and only if it is isomorphic to $A/\mathfrak m$ for some maximal ideal $\mathfrak m$ in $A$.
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 $\mathbf{R}[x]$.
Exercise 111.9.4. Let $A = k[x, y]_{(x, y)}$ be the local ring of the affine plane at the origin. Make any assumption you like about the field $k$. Suppose that $f = x^3 + x^2y^2 + y^{100}$ and $g = y^3 - x^{999}$. What is the length of $A/(f, g)$ as an $A$-module? (Possible way to proceed: think about the ideal that $f$ and $g$ generate in quotients of the form $A/{\mathfrak m}_ A^ n= k[x, y]/(x, y)^ n$ for varying $n$. Try to find $n$ such that $A/(f, g)+{\mathfrak m}_ A^ n \cong A/(f, g)+{\mathfrak m}_ A^{n + 1}$ and use NAK.)
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