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ag.algebraic geometry – Infinite uniform dimension $Rightarrow$ infinitely many idempotents in a localization of a quotient

In common, it isn’t workable to bear such an model $I$ and component $r in R setminus I$ out of your assumption.

An clear obstruction is the category of rings of innumerable uniform dimension having precisely one prime model. If a hoop $R$ has a exclusive prime model, then the identical is undoubted of each nonzero quotient and localization. In specific, $R$ does not have any idempotents, nor does any ring ensuing from $R$ by quotients and localizations.

On the opposite hand, it is simple to bear rings with a exclusive prime model which too have innumerable uniform dimension. You can simply adjoin an innumerable variety of mutually orthogonal nilpotents to a bailiwick, e.g. $R = okay[x_1, x_2, ldots]/ (x_{i}x_j)_{0 leq i leq j}$ accommodates $bigoplus_i x_iR$.

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