Question

Please answer all parts. Thank you!

20. Let R be a commutative ring with identity. We define a multiplicative subset of R to be a subset S such that 1 S and ab S if a, b E S. Define a relation ~ on R × S by (a, s) ~ (a, s') if there exists an s"e S such that s* (s,a-sa,) a. 0. Show that ~ is an equivalence relation on b. Let a/s denote the equivalence class of (a,s) e R x S and let S-R be the set of all equivalence classes with respect to ~. Define the operations of addition and multiplication on S-1R by a bat +bs st s tst respectively. Prove that these operations are well-defined on S-1R and that S-1R is a ring with identity under these operations. The ring S-1Ris called the ring of quotients of R with respect to S. C. Show that the map ψ : R- > S-R defined by ψ(a) a/1 is a ring d. If R has no zero divisors and 0 S, show that is one-to-one. e. Prove that P is a prime ideal of R if and only if S = R is a f. If P is a prime ideal of R andS-R\ P, show that the ring of quotients homomorphism multiplicative subset of R S-1R has a unique maximal ideal. Any ring that has a unique maximal ideal is called a local ring.

Answer 1

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