Notion of the dimension of a ring relative to another ring.

Question

What is the correct notion of the dimension of a ring relative to another ring?

Basically, I'm looking for ways to show that a certain ring is not isomorphic to a subring of another ring. Rings like $\mathbb{Z}$ and $\mathbb{Q}$ intuitively seem one-dimensional and good candidates for measuring the dimensionality of other rings. I'm looking for a way to make that precise that's less naive than the brute force formalism that I came up with.

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I'm trying to come up with a collection of tools for proving that a given ring $S$ is not isomorphic to any subring of $R$. (For proving that $S$ is isomorphic to a subring of $R$, I just construct an explicit injective homomorphism. I don't have any nonconstructive tools for doing this.)

Also, for $S$ to be a subring of $R$, I insist that $1_S = 1_R$. I don't know if this is a standard choice or not.

Anyway, I've been messing around for the most of the morning trying to come up with tests that can show that $\mathbb{Z}[x]$ is not isomorphic to a subring of $\mathbb{Z}$.

So far I have come up with a handful of valid arguments:

1. $\mathbb{Z}$ has no proper subrings. $\mathbb{Z}[x]$ has $\mathbb{Z}[x^2]$ as a proper subring. Thus $\mathbb{Z}[x]$ is not isomorphic to a subring of $\mathbb{Z}$.
2. $\mathbb{Z}$ is additively generated by $1$. $\mathbb{Z}[x]$ is not additively generated by 1.

And a whole mess of invalid arguments. I really like this answer by Jean Marie from which we have the sequence $\mathbb{Z} \subset \mathbb{Z}[x] \subset \mathbb{Q}[x]$ where the first and third rings are PIDs but the middle one is not, which shows that PID-ness is useless for concluding anything about subring relationships.

I came up with one additional argument that appeals to a notion of embedding into what I'm going to call a rewrite ring. If I try to apply this argument to the problem of showing that $\mathbb{Z}[x]$ is not isomorphic to a subring of $\mathbb{Z}$ I might run into circularity issues just because the problem that I'm trying to solve is so simple, but I think it's an interesting idea nonetheless.

I'll define a rewrite ring over $\mathbb{Q}$ of dimension $\kappa$ as $\mathbb{Q}[x_1 \cdots x_\kappa]/I$ where $\kappa$ is a cardinal and $I$ contains $x_1-1$ and for every $i, j$ a term $x_ix_j - l$ where $l$ is a polynomial of degree one. $I$ can additionally contain other stuff. The idea behind $I$ is to guarantee that every element of $\mathbb{Q}[x_1 \cdots x_\kappa]/I$ can be rewritten to a finite formal sum of linear terms of the form $\sum_i k_ix_i$ where $k_i$ is some rational number.

Anyway, $\mathbb{Z}[x]$ has an injective map going to the rewrite ring over $\mathbb{Q}$ of dimension $\omega$, but not the one of dimension one. If it did, then $x$ would have an interpretation as a rational number $p/q$ and would thus satisfy $qx = p$, but $x$ by construction does not satisfy any nontrivial relations with elements of $\mathbb{Z}$.

Also, if I use $\mathbb{R}$, then $\mathbb{Z}[x]$ does have an injective map going to $\mathbb{Z}[\pi] \subset \mathbb{R}$. This suggests to me that $\mathbb{R}$ is not really that great of a yardstick for measuring the dimensionality of other rings unless I impose some other structure on my rings like an some type of order or a topological space.

Answer 1

(All rings are commutative below for simplicity.)

There is simply no such thing as "the correct notion of the dimension of a ring relative to another ring," and such a thing is not necessary for solving your motivating problem anyway. Here are two easy examples: for $f : R \to S$ a ring homomorphism,

• we could consider the "rank" of $S$ over $R$ as a module, e.g. some variation of the minimal number of generators of $S$ as an $R$-module. Already we have various choices for how to define this but a simple one is that in the special case that $R$ is a domain we could consider the dimension of $S \otimes_R Q(R)$ over $Q(R)$ (here $Q$ is the fraction field). For example, if $S = R^n$ this number is $n$.
• we could consider the "rank" of $S$ over $R$ as an algebra, e.g. some variation of the minimal number of generators of $S$ as an $R$-algebra. Again we have various choices for how to define this but a simple one is that if $R$ and $S$ are both domains we could consider the transcendence degree of $Q(R)$ over $Q(S)$. If $R$ is a field this recovers a version of the dimension of a variety. For example, if $S = R[x_1, \dots x_n]$ this number is $n$.
• or we could consider some variation of the Krull dimension, or the Gelfand-Kirillov dimension, or the homological dimension, or...

There is simply no sense in which any of these is "more correct" than the others. They are useful concepts which are relevant in different situations for dealing with different phenomena.

The above ideas can be used to prove that a ring $R$ cannot be a subring of another ring $S$ in various ways, e.g. the following.

1. Given a ring $R$ we can consider its rank as a module over $\mathbb{Z}$, or equivalently $\dim_{\mathbb{Q}} R \otimes \mathbb{Q}$. This number is monotonic in the sense that if $R \hookrightarrow S$ is an inclusion of a subring then $\dim_{\mathbb{Q}} R \otimes \mathbb{Q} \le \dim_{\mathbb{Q}} S \otimes \mathbb{Q}$. For $\mathbb{Z}$ this number is $1$ and for $\mathbb{Z}[x]$ this number is infinite, so the latter can't be a subring (or even an additive subgroup) of the former as desired.

2. Given an integral domain $R$ we can consider the transcendence degree of $Q(R)$ over its prime subfield. This number is also monotonic in the above sense, and for $\mathbb{Z}$ this number is $0$ while for $\mathbb{Z}[x]$ this number is $1$.

Which notions of dimension or rank to use in a given situation just depends on what you're trying to do.