Rings with flat injective envelope and global dimension at most one Is there a classification of rings $R$ with the following properties:
-The injective envelope of $R$ is flat.
-The global dimension of $R$ is at most one.
In case $R$ is a finite dimensional algebra, $R$ is isomorphic to the direct product of rings of upper triangular matrices over a division field up to Morita equivalence, see proposition 1.17 in https://www.sciencedirect.com/science/article/pii/S0001870810000915.
It would be interesting to see other examples of such rings with global dimension one that are not finite dimensional algebras.
 A: If we replace "flat" by "projective", there is a very nice classification that is like the finite dimensional case: see (P1) below.  When the ring is left and right Noetherian, we can give a good classification in the original flat case: see (F3) below. Without chain conditions, the problem seems harder.  Positive examples include HNP rings and countable von Neumann regular rings.
Given a ring $R$, denote the injective hull of $R_R$ by $Q$.
We will say $R$ has property F on the right if $Q_R$ is flat and $R$ has property P on the right if $Q_R$ is projective.  The problem is to classify right hereditary rings with property F or P.  When $R$ is a finite dimensional algebra, the injective hull is finitely generated as a module, so properties F and P are equivalent.
When we use the unadorned term "Noetherian", we mean "right and left Noetherian".  The same convention applies to "Artinian", "hereditary", "Goldie", "property F", etc.  When we say a ring is semisimple, we mean that it is semisimple Artinian, and when we say a ring is regular, we mean that it is von Neumann regular.
Here are some results.  Details are provided below.
(P1) A right hereditary ring $R$ has property P on the right if and only if $R$ is Morita equivalent to a finite direct product of full triangular matrix rings over division rings (equivalently, $R$ is isomorphic to a finite direct product of complete blocked triangular matrix rings over division rings).
(P2) [Corollary of P1] If a right hereditary ring $R$ has property P on the right, then it has property P on the left, and it is hereditary and Artinian, and $Q_R$ and ${}_RQ$ are finitely generated modules.
(FP1) If $R$ is a right hereditary, right Artinian ring, then $R$ has property F on the right if and only if $R$ has property P on the right.
(F1) Every regular ring has property F, since every module is flat.  Note that every regular ring is semihereditary and any regular ring in which all right ideals are countably generated is right hereditary.
(F2) Every commutative semihereditary ring has property F and every semiprime Goldie ring has property F.
A semiprime right Goldie ring that is not left Goldie does not have property F on the right.  Cozzens and Jategoankar, among others, have given examples of principal right ideal domains (so right hereditary, right Noetherian) that are not left Ore.
(F3) Every hereditary Noetherian ring is a finite direct product of hereditary Artinian rings and HNP (hereditary Noetherian prime) rings. By (F2), the HNP rings have property F and so the problem of characterizing hereditary Noetherian rings with property F comes down to characterizing hereditary Artinian rings with property F.  This latter problem is solved by (FP1) and (P1).
(F4) [Corollary of F3] A hereditary Noetherian ring has property F on the right if and only if it has property F on the left.

Our point of view is the following.  For details and references, see the end of this answer. Recall that a ring is right nonsingular if $0$ is the only element whose right annihilator is essential as a right ideal of $R$.  The following rings are always right nonsingular: right semihereditary rings, regular rings, semiprime right Goldie rings, rings with no nonzero nilpotent elements. When $R$ is right nonsingular, the injective hull $Q$ of $R_R$ can be given a ring structure that makes $R$ a subring and agrees with the right $R$-module structure.  With this ring structure, we call $Q$ the maximal right quotient ring of $R$.
Here are some useful facts, assuming that $R$ is right nonsingular.  If $R$ is right semihereditary or $Q$ is semiprime right Goldie or $R$ is right Noetherian, then ${}_RQ$ is flat.  If $Q$ is semisimple, then $Q_R$ is flat if and only if $Q$ is also the maximal left quotient ring of $R$.  If $R$ is a right Goldie ring (e.g., a right Noetherian ring), then $Q$ is always semisimple.
We now address the specific claims above.  We will refer to the books Ring Theory: Nonsingular Rings and Modules by Ken Goodearl and Rings of Quotients: An Introduction to Methods of Ring Theory by Bo Stenström.
(P1)  This is part of Theorem 3.2 from "Generalizations of QF-3 Algebras" by Colby and Rutter, PDF here.
Versions were also apparently discovered independently by Stephenson and Goodearl.  Another reference is [Goodearl, Theorems 5.28 and 5.21].
(P2)  These statements follow from the triangular matrix description in (P1).  The hereditary and Artinian properties also follow from [Goodearl, Theorems 5.21 and 5.23].  The finite generation property follows from [Goodearl, Proposition 5.19] and also from a result of Faith and Walker that says that if the injective hull of a finitely generated module is contained in a direct sum of finitely generated modules, then that injective module is finitely generated.
(FP1) This follows from [Goodearl,Theorem 5.21] or [Stenström, Corollary XII.7.3].
(F1)  This is obvious.  The fact that regular rings with all right ideals countably generated are right hereditary is a consequence of Corollary 2.15 in the book Von Neumann Regular rings by Goodearl.
(F2) When $R$ is right semihereditary, ${}_RQ$ is flat.  Thus for a commutative semihereditary ring, $Q_R$ is flat.
If $R$ is semiprime right and left Goldie, then $Q$ is the Goldie quotient ring on both the right and left.  As left Ore localization is exact, $Q_R$ is flat.
If $R$ is right but not left Goldie, then $Q$ is not the maximal left quotient ring of $R$ and so by the discussion above, $Q_R$ is not flat.  There are several ways to see that $Q$ is not the maximal left quotient ring of $R$; one is that the failure of the left Ore condition implies ${}_RR$ is not an essential submodule of ${}_RQ$.
Here's a simple example of this phenomenon.  Let $D$ be a division ring, $\phi:D\to D$ a ring endomorphism that is not surjective, and $R$ the skew polynomial ring $D[x;\phi]$ with coefficients on the right.  (That is, elements of $R$ have the form $\sum_{i=0}^n x^id_i$ and $dx=x\phi(d)$.)  Every right ideal of $R$ is generated by the monic polynomial of least degree that it contains, so $R$ is a principal right ideal domain.  If $a\in D$ is not in the image of $\phi$, then $xa$ and $x$ have no common left multiple, so $R$ is not left Ore.  We can explicitly see that $Q_R$ is not flat by setting $I=Rx+Rxa$ and noting the multiplication map $Q\otimes_R I\to Q$ is not injective (since $x^{-1}\otimes x-a^{-1}x^{-1}\otimes xa$ is not zero as a tensor but maps to $0$ under the multiplication map).
(F3)  The fact that hereditary Noetherian rings are finite products of hereditary Artinian rings and HNP rings can be found in various sources, such as Theorem 5.4.6 in Noncommutative Noetherian rings by McConnell and Robson.  Statement (F4) follows from this and (F2), (FP1), (P1).
(F4)  This follows from (F3), (FP1), (P2).

This paragraph and the next are intended for those who want to know precise details, with references.
The following rings are always right nonsingular: right semihereditary rings ([Goodearl, Proposition 1.27], [Stenström, Theorem XII.6.4]), regular rings (they are semihereditary), semiprime right Goldie rings ([Goodearl, Corollary 3.32], [Stenström, Lemma II.2.5]), rings with no nonzero nilpotent elements ([Goodearl, Exercise 1.D.3], [Stenström, Lemma XII.5.1]).  A commutative ring is nonsingular if and only if it is semiprime ([Goodearl, Proposition 1.27], [Stenström, examples at the end of §XV.1]).
When necessary, we assume all rings are right nonsingular with maximal right quotient ring $Q$.  A result of Sandomierski shows that a right hereditary ring with finite right Goldie dimension is right Noetherian: see [Goodearl, Corollary 5.20], [Stenström, Exercise XII.9].  If $R$ is right coherent, ${}_RQ$ is flat: see [Goodearl, Corollary 3.7], [Stenström, Example 4 at the end of §XII.2].  Since right semihereditary rings are right coherent, ${}_RQ$ is flat for any right semihereditary ring: see [Goodearl, remark/definition before Corollary 3.7], [Stenström, Example 2 at the end of §I.13].   The ring $Q$ is semisimple if and only $R$ has finite right Goldie dimension: see [Goodearl, Theorem 3.17], [Stenström, Theorem XII.2.5].  When $R$ has finite right Goldie dimension, $Q_R$ is flat if and only if $Q$ is also the maximal left quotient ring of $R$: see [Goodearl, Exercise 3.B.23], [Stenström, Corollary XII.7.3].  When $Q$ is a right Ore localization of $R$, e.g., $R$ is semiprime right Goldie, then ${}_RQ$ is flat, since Ore localization is exact.
A: You should add global dim leq 1. So, k[x] is an infinite dimensional example satisfying both
