Let R = {6 la, be Q}and let p:R – Q be defined by : 0( D = 1) is a ring a) Homomorphism. b) Isomorphism c) None 2) ker (4) is generated by ro a) 8 1 b) I J 3) R/ker (4) = a) Z b) Q c) R d) None 4) Ker (4) is: a) Prime ideal not maximal b) Maximal ideal not prime Both maximal and prime d) None

Q: Show that f:Z2 -> Z2 defined by f(n) = n2-n is a ring homomarphism.

A: The element in Z2 are {0,1} Now, f(0) = 02 - 0 = 0           f(1) = 12 - 1 = 0 Hence, f maps Z2 to…

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

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Q: Let R be a ring and define R* = ∆R = {(r, r) : r ∈R }.1 (a) Suppose R = Z6 and list the members of…

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Q: Let T be ring containing elements e, f, both # 0T, such that e + f = 1r , e² = e, f² = f , and e · f…

A: We are given e+f = 1T,e2=e,f2=f and e.f=0T where T is the rings containing elements e,f both not…

Q: a) Let R and S be any rings. Show that the zero map, (ie, f: RS defined by f(r) = 0s for all rER) is…

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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: 4) Let R be a ring and F-M₂(R) be ring of 2 x 2 matrices over R under standard addition and…

A: # Answer 4) Given:- R  is a ring.- F=M2​(R) , the ring of 2×2  matrices over R  under standard…

Q: Prove that the extension of scalars from Z to the Gaussian integers Z[i] of the ring R is isomorphic…

A: It is given that ℤi is the Gaussian ring of integers. The objective is to prove that ℤi⊗ℤℝ≅ℂ. To…

Q: 5. Prove that the intersection of any set of ideals f a ring is an ideal. Hint: Let T 1, be an…

A: In the given question we have to prove that  the intersection of arbitrary number of ideals is again…

Q: 5) Let (R,+) and (R.+,) be any two rings, let f: R R be a ring homomorphism Ker () (0), then a) f is…

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Q: 3. Consider the function f: ZQ defined by f(n) = n. a) Prove f is a ring homomorphism. b) Find the…

A: Step 1: Step 2: Step 3: Step 4: