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Recall that a commutative ring is Jacobson if every prime ideal is the intersection of the maximal ideals that contain it. In the exercises of a commutative algebra course I gave I asked the student …

The answers to the question prime ideals in C([0,1]) explain why the answer is no, assuming that Yemon Choi was mistaken in his answer in believing that you are only interesting in continuous homomorp …

Suppose that $R$ is a complete DVR with field of fractions $K$, uniformiser $\pi$ and residue field $k$. Let $B$ be a subring of the ring $K(t)$ of rational functions over $K$. Moreover assume that $ …

I think that the question difficult as illustrated by Hailong's answer. I suspect that it will be hard to even find a nice parameterisation of the $H$-orbits of maximal ideals in your refined question …

In case anyone else has the same question and discovers this page I have just found a more explicit reference for this result: Remark 3.4.4 of A Singular introduction to commutative algebra by Greuel …

In non-commutative algebra Krull dimension has been generalised by Gabriel & Rentschler. A decent account of it can be found in Chapter 6 of McConnell and Robson's book on non-commutative Noetherian r …

Often the most useful dimension in non-commutative algebra is the length of the minimal injective resolution of the ring as a module over itself. In many important cases this is the same as the global …

Suppose that $M$ is a finitely generated module over $A=k[X_1,\ldots,X_n]$ of Krull dimension $m$ with $k$ an infinite field. Then one version of Noether normalisation says there is an $m$-dimensional …

Example 1.2.2 in Chapter 1 of Weibel's book in progress on K-theory http://www.math.rutgers.edu/~weibel/Kbook.html says that $R_2$ in the notation of your question has a stably free module that is not …