5. Let R be a ring and r1,., r, € R. Prove that the subset (r1,., "n) = {Airı + Anrn | A1,. An E R} .... .. ..... is an ideal in R.

Q: Problem 11. Let F(R, R) be the vector space over R of all functions f: RR. Prove that if a, b e R…

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Q: 4. Let R be a ring. (a) Let x, y, z € R. Prove that, if x | y and y | z, then x | z. (b) Prove that…

A: Let R be  a  ring .4) a)  Let  Suppose that   divides y and  y  divides z.We  have  to show that x…

Q: Recall that {C, +, ·} is the field of all complex numbers. Introduce a total order < on C as…

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Q: Let 2Z and 5Z be ideals of Z. Compute 2Zn5z. Prove that 1 E 2Z+5Z.

A: We use basic concepts of ideals to solve this problem.

Q: Is there a ring that has exactly one proper ideal?

A: We have to find a ring that has exactly one proper ideal.

Q: 4. Let p: R S be a ring homomorphism. Show that J = ker p is a prime ideal if S is a domain. Show…

A: Fundamental theorem of homomorphism: Let R and S be rings. Consider the homomorphism φ:R →S. Then,…

Q: d) None 4) Ker (6) is: a) Prime ideal not maximal b) Maximal ideal not prime c) Both maximal and…

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Q: 5. In the ring (Mat,x2(Z), +,.), if A € Matzx2(Z) then the ideal (A) = (a) {B. A: BE Matzx2(Z)} (b)…

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Q: a b r s - Let R а, b, d 0 d r, s, tE Z, s and S even}.ir If S is an ideal of R, what can you say…

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