Test2F_07 - an ideal in R x S(b Let I be a maximal ideal in...

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MAT 305 Test 2 Fall’07 I. List all the units and in each case find its multiplicative inverse in : (a) Z 8 , (b) Z [i] , (c) Z 7 [x] , (d) Q ( 2) II. Find all the zero divisors ( if any ) in : (a) Z 7 [x] , (b) Z 6 [x] , (c) Z 2 x Z 3 , (d) Z [i] III . Find all ring homomorphisms ; (Justify your answers) (a) ϕ : Z 8 Z 10 , (b) ϕ : Z 5 Z 10 , (c) ϕ : Q Q , (d) ϕ : Q ( 5) Q ( 5) IV . (a) Let I be an ideal in a ring R and J an ideal in a ring S. Show that I x J is
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Unformatted text preview: an ideal in R x S. (b) Let I be a maximal ideal in a commutative ring R with unity and S any ring. Show that I x S is a maximal ideal in R x S. V. Prove one of the following : (a) If D is an integral domain, then for all non zero polynomials f(x), g(x) in D[x], deg(f(x)g(x)) = deg f(x) + deg g(x) (b) If F is a field then {0} and F are the only ideals in F....
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