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

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

(i) Show that S defined as R \ I (the complement of I in R) is multiplicatively 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 is isomorphic to the

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

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

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Expert Solution

Colby Messinger answered 1 year ago

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