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  3. let r be a commutative domain, and let i be a prim

Let R be a commutative domain, and let I be a prime ideal of R. (i) Show...

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Advance Math

Let R be a commutative domain, and let I be a prime ideal ofR.

(i) Show that S defined as R \ I (the complement of I in R) ismultiplicatively closed.

(ii) By (i), we can construct the ring R1 =S-1R, as in the course. Let D = R / I. Show that

the ideal of R1 generated by I, that is,IR1, is maximal, and R1 / I1R isisomorphic to the

field of fractions of D. (Hint: use the fact that everything inS-1R can be written in the

form s-1r, where s ∈ S and r ∈ R. The first step isto show that IR1 ∩ R = I).

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