By Thomas C. Craven

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**Extra resources for Abstract algebra [Lecture notes]**

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2. In Z6 , the set I = { [2k] ∈ Z6 | k ∈ Z } is an ideal. 3. p(x)R[x] = { p(x)f (x) | f (x) ∈ R[x] } is an ideal of R[x] for any commutative ring R with 1. 4. In Z[x], the set I = { f (x) ∈ Z[x] | f (0) ≡ 0 (mod n) } is an ideal for any n ≥ 2 in Z. This generalizes an example on page 136 where n = 2. 5. For R = C(R, R), fix any r ∈ R. The set I = { f ∈ R | f (r) = 0 } is an ideal. Note that it does not work to use a number other than 0. 1 2 a b a, b ∈ R is a right ideal but not 0 0 a left ideal.

8 Example: in the group G = Z2 × Z3 × Z5 , the elements (1, 0, 0), (0, 1, 0) generate a subgroup of order 6, isomorphic to Z2 × Z3 . , (n, n, n) = (0, 0, 0) only if n is a multiple of 30), so G = (1, 1, 1) ∼ = Z30 . Exercise 9, page 188: (a) choose any a ∈ / Z(G). Then b = a−1 gives a counterexample. (b) ab ∈ Z(G) implies that it commutes with a−1 , in particular. Thus, b = a−1 ab = aba−1 ; multiplying by a on the right gives ba = ab. This is typical of proving things about groups: you have to find the right element to apply things to.

15) 3. Let F = Q( 2) = { a + b 2 √ | a, b ∈ Q }√be the field you saw on homework. Define a function f : F → F by f (a + b 2) = a − b 2. Show that f is an isomorphism. (15) 4. Define integral domain. Give three examples, at most one of which is a field. (20) 5. Let f : Z20 → Z5 be the ring homomorphism defined by f ([n]20) = [n]5 . a. Find K = { x ∈ Z20 | f (x) = 0 }. b. Show that K is a subring of Z20 . c. Check the things which apply: f is injective ; f is surjective Z20 is a commutative ring , integral domain Z5 is a commutative ring , integral domain , field , field (15) 6.