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I had asked this question in one of the seminars I attend and they said I should ask it here. Let R be a commutative ring with unity where $1 \neq 0$ and suppose that R is not Noetherian (i.e. all of the 3 equivalent conditions: max.c, acc, and finitely generated ideals - all fail concurrently). Can we take this R through some sort of algorithim to obtain a similar ring R(N) which is Noetherian? I get that the usage of "similar" here is really vague but I am not sure how else to phrase it. Also I would like for this new ring R(N) to be commutative with unity, and having $1 \neq 0$ ideally (or at least left and right Noetherian if it is not commutative let's say). If there is really nothing of this sort, then my question becomes, what is the next best thing? Again that is vague but not sure how else to phrase it. If there is such a thing, then can we generalize it to modules over a commutative unital ring with $1 \neq 0$?

(I tried googling this, no luck there).

Countable
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1 Answers1

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Let $\newcommand{\CRing}{\mathrm{CRing}}\CRing$ be the category of commutative, unital rings and let $\CRing_N$ be the full subcategory of Noetherian rings. Usually something like a "Noetherialization" would mean an adjoint to the inclusion of categories, so we can ask whether it is a reflective subcategory (which means that the inclusion $\CRing_N\hookrightarrow\CRing$ has a left adjoint). This however fails since this functor does not create limits. For instance, a countable product of the ring $\mathbb Z$ is not Noetherian, so it cannot be reflected to the subcategory of Noetherian rings.

Considering colimits should similarly show that this category is not coreflective either, so inclusion doesn't admit a right adjoint either.

Wojowu
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