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# Rings of Unity, Monoid, Momomorphism and Invertible Elements

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Let R commutative ring with unity, and S a sub monoid of the multiplicative monoid of R. In RxS define (a,b) ~ (b,t) if &#1069;u &#1108; S &#1101; u(at-bs)= 0. Show that ~ is an equivalence relation in RxS. Denote the equivalence class of (a,s) as a/s and the quotient set consisting of these classes as RS-¹. Show that RS-¹ becomes a ring with unity relative to
a/b +b/t = (at +bs)/st
(a/s)(b/t) = (ab)(st)
Multiplicative unity = e/e

Show that a --> a/e is homomorphism of R into RS&#713;¹ and this is a monomorphism if and only if no element of S is a zero divisor in R.

Show that the elements s/e, s &#1028; S, are units (set of invertible elements) in RS-¹.

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Proof:
First, I show that ~ is an equivalent relation in . From the definition, we know that if and only if we can find some , such that . We need to verify the following three conditions.
1) Reflexive: Because is commutative, then . So for any , we have . Thus .
2) Symmetric: . Since , we can find some , such that . This implies ...

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