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Question 5 (6 points) Let o: R+S be a ring homomorphism. Suppose that o(R) and ker...
18. Let o: R+ S be a ring homomorphism. Prove each of the following statements. (a) If R is a commutative ring, then (R) is a commutative ring. (b) (0)=0. (c) Let 18 and 1s be the identities for R and S, respectively. If o is onto, then (1r) = 1s. (d) If R is a field and $(R) +0, then (R) is a field.
5. Let I be an ideal in a ring R. Prove that the natural ring homomorphism T: RRI has kernel equal to I.
(3.) (a) Suppose that y: R S is a ring homomorphism. Please prove that (-a) = -f(a) for all a ER (b) Suppose R and S are rings. Define the zero function y: R S by pa) = Os for all GER. Is y a ring homomorphism? Please explain. (4.) Suppose that p is a prime number and 4: Z, Z, is defined by wa) = a.
Please solve all questions 1. Let 0 : Z/9Z+Z/12Z be the map 6(x + 9Z) = 4.+ 12Z (a) Prove that o is a ring homomorphism. Note: You must first show that o is well-defined (b) Is o injective? explain (c) Is o surjective? explain 2. In Z, let I = (3) and J = (18). Show that the group I/J is isomorphic to the group Z6 but that the ring I/J is not ring-isomorphic to the ring Z6. 3....
PLEASE do parts a, b, and c. Thank you. Let f: R S be a homomorphism of rings. Let J be an ideal in S. Let I = {r E R : f(r) € J}. la Prove: ker f +$. « Prove: ker f SI. Prove: I is an ideal in R.
QUESTION 4 (a) Let RS be a ring homomorphism with I an ideal of R and J an ideal of S. Define 0(I) = {$(1) I ET) and o-'(J) = {ve R(y) € J} and check as to whether or not (i) °(1) is an ideal of S (6) (ii) o-'() is an ideal of R (6) (Hint: I, J are two-sided ideals and in both cases of (i) and (ii) above, first check the subring conditions) (b) Given a...
= a (mod n) is a ring homomorphism. (10) Suppose that o Z Z defined by ¢(a) (a) (5 Pts) Prove that o is injective. Answer (b) (5 Pts) Prove that o is surjective onto its image. Answer = a (mod n) is a ring homomorphism. (10) Suppose that o Z Z defined by ¢(a) (a) (5 Pts) Prove that o is injective. Answer (b) (5 Pts) Prove that o is surjective onto its image. Answer
3. Let y: K + Aut(H) be a homomorphism. Write (k) = Ok. Let G be a group. A function d: K + H is called a derivation if dikk') = d(k) (d(k')). Show that d: K + H is a derivation if and only if V: K + H y K given by v(k) = (d(k), k) is a homomorphism. 4. Suppose that a: G + K is a surjective homomorphism and that 0: K + G is a...
4. An element a in a ring R is called nilpotent if there exists a non-negative integer n such that a" = OR (a) Let a and m > O be integers such that if any prime integer p divides m then pſa. Prove that a is nilpotent in Zm. (b) Let N be the collection of all nilpotent elements of a ring R. Prove that N is an ideal of R. (c) Prove that the only nilpotent element in...
Let R be a ring, let S be a subring of R and let' be an ideal of R. Note that I have proved that (5+1)/1 = {5 +1 | 5 € S) and I defined $:(5+1) ► S(SO ) by the formula: 0/5 + 1)=5+(SNI). In the previous video I showed that was well-defined. Now show that is a ring homomorphism. In other words, show that preserves both ring addition and ring multiplication. Then turn your work into this...