Let $R$ be a ring, and $M$ an $R$-module. What is the definition of a filtration of $M$? Is it simply a descending sequence of submodules $M_{1} \supset M_{2} \supset M_{3} \supset \dots \supset \dots$. Does it have to be countably infinite or can it be finite? For example, what is an example of a filtration of $M = \mathbb{C}[x,y] / \langle x^{2}y \rangle$ as a $\mathbb{C}[x,y]$ module.
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I don't think there are usually any restrictions on the number of terms of the sequence. There are descending filtrations like the one you wrote, or ascending filtrations $M_1\subset M_2\subset M_3\subset \cdots$. Sometimes the number of terms are left ambiguous. Here's an example of an ascending filtration on the module you asked: $(0)\subset (x) \subset M$. Here's another but, descending: $M\supset (0) \supset (0)\supset \cdots$. – Eoin Feb 26 '20 at 22:04
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does the chain have to properly decrease/increase? – user100101212 Feb 26 '20 at 22:13
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Not usually; if this was required the author would probably mention it as a convention. – Eoin Feb 26 '20 at 22:14
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Does the sequence have to terminate with the zero module? – user100101212 Feb 26 '20 at 23:12
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No, that's not required. – Eoin Feb 26 '20 at 23:16