Let S = {: la, be R}and let o:S R be defined by: = a . 1) o is a ring a) Homomorphism. b) Isomorphism c) None 2) ker (4) is generated by a) 01 b) : ] 3) S/ker (4) = a) z b) Q c) R d) None

Q: Let R be a ring and consider R x Z = { (r,n) | r e R, ne Z} Define (r,n) + (s,m) =(rts, n+ m) (r,…

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Q: Let R = {6 la, be Q}and let p:R – Q be defined by : 0( D = 1) is a ring a) Homomorphism. b)…

A: Here, first we will check the the isomorphism property and then we will find the kernel , See the…

Q: R = {[ la.b e z}and let p:R -- Z be definad by : o(l; ) = - %3D 1) o is a ring a) Homomorphism. b)…

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Q: 4) Let f : C – C be a homomorphism of rings with f(x) = x for all x E R. Show that f is either the…

A: If a function f is ring homomorphism from a ring R to another ring R'. Then We knew from the…

Q: ring R is commutatve a) Let R be a %3D Then prove that

A: (NOTE- These are two different questions, only required to do the first one. Please post the second…

Q: = a . Let S = { la, b e R} and let p:S → R be defined by : Ø( D 1) is a ring

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Q: a) x+1 b) x² +1 c) x d) x 2- The value that replacesx is a) x+1 b) x?+1 c) X d) x? 3- (x+1) is equal…

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Q: Let R and S be rings.a. Show that the mapping from R ⨁ S onto R given by (a, b) --> a is a ring…

A: Since you have posted multiple questions, we will provide the solution only to the first question as…

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) f a²+ a ECCR) for all a E R, then Poove thot R is Commutalve.

A: According to the company's guideline, we shall prove first Que only. For others, kindly post again…

Q: Let f:R, » R, and let g : R, » R, be two ring isomorphisms. If ø =g•f, then 2- ø(e,) = a) , b) € c)…

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Q: Let R = { la, b e Z}and let o:R- Z be defined by : 1) ois a ring a) Homomorphism. b) Isomorphism c)…

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Q: 1- Let ý:R, » R, be a ring homomorphism such that Kerø = . Then, ø is a) 1-1 b) onto c) Both 1-I and…

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Q: = {I6 Tla, b e Q}and let p:R - Q be defined by : 0( 1) ois a ring a) Homomorphism. b) Isomorphism c)…

A: Since you have asked multiple question, we will solve the first question for you. If you want any…

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: Let R = %3 { la, b e z}and let p:R - Zbe defined by : 0(1 ) = - . 1) is a ring a) Homomorphism. b)…

A: The solution is given by using definitions of homomorphism, isomorphism and kernel as follows

Q: Q bo dofined by : 0ED= Let R 1) ois a ring a) Homomorphism. b) Isomorphism None c) 2) ker(6) is…

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Q: {E Jla, b e R} and let :S –R be defined by : 0( ) = 1) ois a ring a) Homomorphism. b) Isomorphism c)…

A: Using conditions of homomorphism of rings  and  isomorphism theorem we can answer this questions

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: Exercise: Prove that the homomorphism f:R R of the ring R onto the ring R' is isomorphism if and…

A: We have given that , f : R → R' be homomorphism. Also we have given that ,  f : R → R'  is onto. We…

Q: Let R, S, T be rings and let 1: RS and 92: ST be ring homomorphisms. Show that: (a) ₁0₁: R→T is a…

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Q: 6. Let R be a commutative ring with identity and let f be an onto homomorphism from R to a ring S.…

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Q: 8. If R is a field, then: a) = R[x] b) is prime c) is maximal d) None of above

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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: 1. let S: :1 xg*Z a) Show that is a ning. ning. b) Show that ¢: S Z olefined by $ ( [Š 87)-* is a…

A: Solution of part(a): It is given that S=x0y0, x,y∈ℤ. We know that the sum and product of two 2×2…

Q: 1+xr 1+x* a) b)

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Q: {: Tla, b e R} and let p:S -R be defined by : 0(; ) = c Let S = %3D 1) is a ring a) Homomorphism. b)…

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Q: Let S = { la, b e R} and let p : s –R be defined by : ø( ) = c 1) is a ring a) Homomorphism. b)…

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