Lemma 10.31.4. Let R be a Noetherian ring.
Any finite R-module is of finite presentation.
Any submodule of a finite R-module is finite.
Any finite type R-algebra is of finite presentation over R.
Lemma 10.31.4. Let R be a Noetherian ring.
Any finite R-module is of finite presentation.
Any submodule of a finite R-module is finite.
Any finite type R-algebra is of finite presentation over R.
Proof. Let M be a finite R-module. By Lemma 10.5.4 we can find a finite filtration of M whose successive quotients are of the form R/I. Since any ideal is finitely generated, each of the quotients R/I is finitely presented. Hence M is finitely presented by Lemma 10.5.3. This proves (1).
Let N \subset M be a submodule. As M is finite, the quotient M/N is finite. Thus M/N is of finite presentation by part (1). Thus we see that N is finite by Lemma 10.5.3 part (5). This proves part (2).
To see (3) note that any ideal of R[x_1, \ldots , x_ n] is finitely generated by Lemma 10.31.1. \square
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