Let R be a regular ring with 1. (i) Prove that for any a E R, there exists an idempotent e ER such that Ra Re. -- (ii) Prove that for any two idempotents e, f € R, there exists an idem- potent g E R such that Re + Rf = Rg.

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: Let R be a ring in which each element is idempotent. Let R = R XZ₂. Define + and x on R by (a, [n])…

A: Given: R is a ring in which each element is idempotent and R=R×Z2.To show: + and × are well-defined…

Q: 4. Let R be a nontrivial ring. Prove that R is an integral domain if and only if the following…

A: To prove that a nontrivial ring R is an integral domain if and only if the cancellation property…

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: 1. Let R be a ring with the additive identity 0. Prove that for any a E R, 0- a = 0.

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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: 3 Let I be an ideal of the ring R and define the set ann (1) ={rERIra=o, VaE Prove that ann (I)…

A: We have to show that  ann(I) is left as well as right ideal of R

Q: Let f: R → S be a ring homomorphism and T ≤ S. Prove that K = { r ∈ R : f (r) = 0s} and P = { r ∈ R…

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Q: Let R′ be a commutative ring of characteristic 2. Show that                    I ={r∈R′|r^2 =0} is…

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Q: Prove That Let X and Y be two fuzzy rings of R and R' respectively and f: R R is a homo. Then 1- if…

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Q: Let R be a unital ring. Let r e R be a nilpotent element. Show that 1R- r e Inv (R). (6) Let 4 E R…

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