Localization in the D+M construction. Let T be an integral domain, and let K be a subfield of T and M a maximal ideal of T such that T=K+M. Let D⊆K be an integral domain, and set R=D+M. My question is: if m∈M such that m≠0,is Rm=Tm? In this context, Rm is the localization of R at the multiplicative set S={mk∣k∈N∪{0}}.

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Localization in the D+M construction.  Let T be an integral domain, and let K be a subfield of T and M a maximal ideal of T such that T=K+M. Let D⊆K be an integral domain, and set R=D+M. My question is: if m∈M such that m≠0,is Rm=Tm?  In this context, Rm is the localization of R at the multiplicative set S={mk∣k∈N∪{0}}.

lapinar7g1 · Accepted Answer

The answer is yes. Since R⊆T,also Rm⊆Tm; for the other direction, it suffices to show that T⊆Rm, so fix any a∈T. Then am∈M, since M is an ideal of T and m∈M. Hence, since M⊆R, we have am∈R. Thus a=amm∈Rm, as desired.

Localization in the D+M construction. Let T be an integral domain, and let K be a subfield

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