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This old MO thread and its comments contains a discussion of the Zariski density proof of Cayley-Hamilton (I have also asked a separate question about the proof Victor gives in the comments here). Vic …

Many commutative algebra textbooks establish that every ideal of a ring is contained in a maximal ideal by appealing to Zorn's lemma, which I dislike on grounds of non-constructivity. For Noetherian …

Here is a concrete example which may be useful to think about. It is not exactly the situation you describe but is similar. Let $k$ be a field of characteristic zero and let $R = k[x, y]/(y^2 - x^3 - …

Here is a different proof which maybe clarifies a different aspect of the situation. The ring $C(X)$ of continuous functions on a compact Hausdorff space, as an abstract ring, actually knows its $C^{\ …

Some commentary on David's very nice answer that might provide some useful context. If $D$ is a subring of $\mathbb{R}$ then it must be an integral domain and its field of fractions $\operatorname{Fra …

Here is an alternate approach, more algebraic and less geometric. As in Noam's answer we'll consider the more general equation $x^2 - Dy^2 \equiv 1 \bmod p$. Consider the $\mathbb{F}_p$-algebra $A = \ …

Writing $\mathbb{R}/\mathbb{Z} \cong \mathbb{Q}/\mathbb{Z} \oplus \bigoplus_I \mathbb{Q}$ where $I$ indexes a Hamel basis for $\mathbb{R}$ minus one element, we have $$\text{Ext}^1(\mathbb{R}/\mathbb{ …

I know of three ways to define the dimension of a finitely-generated commutative algebra A over a field F: The Gelfand-Kirillov (GK) dimension, based on the growth of the Hilbert function. The Krull …

Sure, we can define such things. Let's work in the Morita 2-category $\text{Mor}(k)$ over a commutative ring $k$, which has objects $k$-algebras $A$, morphisms $k$-bimodules (with composition given …

The set of even numbers is a non-unital ring, in particular it has an associative multiplication, but you are right that it isn't an algebra over a field. For an example of a non-unital algebra, cons …

Recently I realized that the only PIDs I know how to write down that aren't fields are $\mathbb{Z}, F[x]$ for $F$ a field, integral closures of these in finite extensions of their fraction fields that …

Fernando and Pierre-Yves in the comments are right; $R$ has this property (the version where the canonical map is an isomorphism, as YCor says in the comments) iff it is a solid ring, meaning the mult …

More precisely, how does one characterize integrally closed finitely generated domains (say, over C) based on geometric properties of their varieties? Given a finitely generated domain A and its inte …

The conceptual point is that all of these invariants are Morita invariant because they can be defined directly in terms of the category of modules. Explicitly: Starting from the category of modules …

This is impossible by the Mason-Stothers theorem (which holds over any algebraically closed field of characteristic zero). We want to find $f, g, h$ such that $f + g = h$ where $g$ is a constant and …