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Rings and Subrings

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1. Ler R be a ring, and , prove, using axioms for a ring, the following

? The identity element of R s unique
? That -r is the unique element of R such tht (-r)+r = 0.

(hint, for part 1, suppose that 1 and 1' ate two identities of R, show that 1-1' must be zero, and for part 2, suppose that there is an element such that s+r = 0, and prove that s = -r)

2. let R be the set of complex 4th roots of 1. so R = {1,-1,i,-i} . Does R, together with the usual addition and multiplication of complex numbers, form a ring? Justify your answer.

3.
? Let R be the ring . Show that is a subring but not an ideal of R.
? Let R be a ring. Define what is meant by a polynomial over R in the inderminate x.

4. let and be polynomials in . Calculate f + g and fg in .

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Rings and Subrings are investigated. The solution is detailed and well presented.

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1. Proof:
(a) Suppose and are two identities of . Then for any , we have and . Then . Especially, when , we have . This implies . Therefore, the identity of is unique.
(b) Suppose there is another element , such that ...

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