I am studying commutative algebra at the moment, so all rings are assumed commutative (and unital).
Does there exist a Jacobson local ring $\newcommand{\mfm}{\mathfrak{m}}(A, \mfm)$ that is not an Artinian ring?
For the context I've been studying in, Jacobson rings are defined as rings where all prime ideals are intersections of maximal ideals. A local ring is a ring containing exactly one maximal ideal.
I have come across the result that a ring is Artinian if and only if it is Noetherian and every prime ideal is maximal. Since $A$ is Jacobson, we can deduce that $\mfm$ is the unique prime ideal of $A$, so in particular every prime ideal of $A$ is maximal. My question should then be equivalent to finding a Jacobson local ring that is not Noetherian.
Let $I = (x_ix_j ~|~ i,j \in \mathbb N) \subset \mathbb Q[x_0,x_1,x_2, \dotsc] =: R$. Then $R/I$ is Jacobson and local because every prime ideal contains $x_i^2$ thus contains $x_i$, i.e. is equal to the maximal ideal $(x_0,x_1, x_2, \dotsc)$.
But it is not noetherian, because the maximal ideal is not finitely generated. Hence it is also not artinian.