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What's the easiest example of a non-coherent ring ?

I know that the ring of polynomials in an infinite number of variables over a Noetherian ring A is an example of a coherent ring, so possibly if we take A non-noetherian we can find a non-coehrent ring. but it would be 'looking' really complicated.
So is there an example of a non-coeherent ring that is easy to undersatand and check?

Laassila souhayl
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  • Vakil gives the example of the ring of germs of smooth functions at a point $0 \in \mathbb{R}$: http://math.stanford.edu/~vakil/216blog/incoherent.pdf – Qiaochu Yuan Sep 29 '20 at 18:32

2 Answers2

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Let $V$ be a countable dimensional $F$ vector space, and consider the trivial extension $R=k\times V$ where addition is coordinatewise and multiplication is $(a,v)(b,w)=(ab, aw+bv)$.

This is a commutative, local, perfect, non-Artinian ring, with a nil radical, and it is known that a commutative, perfect ring is Artinian iff it is coherent. (See for example Lam's Lectures on Modules and Rings, exercise 2 p. 161.)

rschwieb
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  • You may also have some luck reasoning that a simple submodule $N$ of ${0}\times V$ is not finitely presented: personally I lack the confidence to reason how it works for a general homomorphism $R^k\to N$. Obviously it's true when $k=1$ because the kernel is clearly all of $V$. – rschwieb Sep 29 '20 at 19:54
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Recall that a ring is coherent iff both (1) intersections of f.g. ideals are f.g. and (2) annihilators of f.g. ideals are f.g.

Either of these criteria gives pointers to constructing non-coherent rings with ease.

For example, to force (1) to fail we might construct an example like $D[x, y, w_i, z_i]/(xz_i - yw_i)$, in which the intersection $(x) \cap (y)$ is not finitely generated.

Or to force (2) to fail, we might consider an infinite product of rings $R = \prod_{i=0}^{\infty} R_\alpha$, and then look at the subring $R'$ generated by the unity and the elements of finite support. The element $(1, 0,0,0, \ldots)$ is annihilated by every element of $R'$ which is $0$ on the first coordinate, which is not an f.g. ideal.

Badam Baplan
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