Let $k$ be a commutative ring. Is there a name for those commutative $k$-algebras with the property that every subalgebra is finitely generated? (Equivalently, the partial order of subalgebras is Noetherian.) Can we say something about their structure? Where can I read more about them? I am particularly interested in the cases $k=\mathbb{Z}$ and when $k$ is field.


Notice that if $k$ is a field then $k[x,y]$ [does not](https://mathoverflow.net/questions/48798) have the property, but $k[x]$ [does](http://www.ams.org/journals/proc/1957-008-05/S0002-9939-1957-0091273-0/S0002-9939-1957-0091273-0.pdf). One can also check directly that the subrings of $\mathbb{Z} \times \mathbb{Z}$ are finitely generated because they are given by $\mathbb{Z}[(n,0)] = \mathbb{Z} \times\_{\mathbb{Z}/n} \mathbb{Z}$, where $n \geq 0$. I wonder if there is a more conceptual reason for this. 


(Usually an object is called Noetherian if its partial order of subobjects is Noetherian, and this applies for example to (non-abelian) groups and modules. But a *ring* is usually called Noetherian if its partial order of *quotients* is Noetherian. This is somewhat confusing. Therefore, although it would be consistent to call a ring Noetherian if its partial order of subrings is Noetherian, this would contradict the usual terminology.)


**Edit.** If $k$ is a field, then a necessary condition is that the algebra is finitely generated of Krull dimension $\leq 1$. This follows directly from Noether normalization and the fact, already mentioned above, that $k[x,y]$ does not have the property. But this condition is not sufficient, as the example $k[x,y]/(x^2)$ shows.


**Summary of the answers so far.** Keith Kearnes suggests the terms "supernoetherian" (if $k$ is Noetherian) and "hereditary finitely generated" (which sounds very good). YCor has reduced the general classification to the case of finitely generated $k$-domains (if $k$ is Noetherian). The classification in this case is still open. If $k$ is a field, is it equivalent to Krull dimension $\leq 1$? Is there a characterization if $k$ is not a field?


YCor has also generalized my observation on $\mathbb{Z} \times \mathbb{Z}$: The $k$-algebra $k \times k$ is hereditarily finitely generated if and only if $k$ is Noetherian, because in fact there is an isomorphism of partial orders between ideals of $k$ and subalgebras of $k \times k$ given by $I \mapsto k \cdot (1,1) + I \times \{0\}$.



Algebras whose subalgebras are finitely generated
**Let $k$ be a commutative ring. Is there a name for those commutative $k$-algebras with the property that every subalgebra is finitely generated? $\ldots$ Where can I read more about them?**


The paper



> 
> Rogalski, D.; Sierra, S. J.; Stafford, J. T.,
> Algebras in which every subalgebra is Noetherian. Proc. Amer. Math. Soc. 142 (2014), no. 9, 2983-2990. 
> 


introduces the term *supernoetherian* for a not-necessarily-commutative $k$-algebra $A$ that has the property that all subalgebras of $A$ are both (i) finitely generated and (ii) Noetherian. In the commutative case, when $k$ is Noetherian, (ii) follows from (i) by the Hilbert Basis Theorem, so these are exactly the $k$-algebras asked about here when $k$ is Noetherian. The authors of this paper consider only the case where $k$ is an algebraically closed field, and in this case they do observe that the commutative supernoetherian algebras have Krull dimension at most $1$, but they do not classify them.


