I study mainly modules and abelian groups, but even I find the Noetherian property useful. For me, probably the main feature lurking in the background is that submodules of a finitely generated $R$-module, $R$ Noetherian, are finitely generated. This fact also allows us to conclude that a finitely generated module over a Noetherian ring $R$ is finitely presented. For a finitely generated module over an arbitrary ring, we always have the exact sequence $$0 \rightarrow K \rightarrow R^n \rightarrow M \rightarrow 0$$ where $K$ is the kernel of the map $R^n \rightarrow M$ that sends the standard basis elements of $R^n$ to the generators of $M$. We say $M$ is finitely presented if $K$ is finitely generated. If $R$ is Noetherian, this is automatic.

It is also nice that when $R$ is Noetherian, any finitele generated $R$-module has a primary decomposition.

Facts like this are standard fare in commutative algebra. Now it has been a good while since I've done much algebraic geometry, but if memory serves, when decomposing an affine variety into irreducible subvarieties, one uses a primary decomposition of the ideal$I$ of the variety (in the polynomial ring in $n$ variables over the field underlying the affine space; this field is usually assumed algebraically closed). But the ideal $I$ turns out to be equal to its own radical, so the primary decomposition involves only prime ideals, and each of these corresponds to an irreducible (affine) variety, each being a component of the original variety.

I study mainly modules and abelian groups, but even I find the Noetherian property useful. For me, probably the main feature lurking in the background is that submodules of a finitely generated $R$-module, $R$ Noetherian, are finitely generated. This fact also allows us to conclude that a finitely generated module over a Noetherian ring $R$ is finitely presented. For a finitely generated module over an arbitrary ring, we always have the exact sequence $$0 \rightarrow K \rightarrow R^n \rightarrow M \rightarrow 0$$ where $K$ is the kernel of the map $R^n \rightarrow M$ that sends the standard basis elements of $R^n$ to the generators of $M$. We say $M$ is finitely presented if $K$ is finitely generated. If $R$ is Noetherian, this is automatic.

It is also nice that when $R$ is Noetherian, any finitely generated $R$-module has a primary decomposition.

Facts like this are standard fare in commutative algebra. Now it has been a good while since I've done much algebraic geometry, but if memory serves, when decomposing an affine variety into irreducible subvarieties, one uses a primary decomposition of the ideal$I$ of the variety (in the polynomial ring in $n$ variables over the field underlying the affine space; this field is usually assumed algebraically closed). But the ideal $I$ turns out to be equal to its own radical, so the primary decomposition involves only prime ideals, and each of these corresponds to an irreducible (affine) variety, each being a component of the original variety.