Show that the union of a chain I1 ⊂ I2 ⊂ .... of ideals of a ring R isan ideal of R.

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: Consider the ring R = 3Z and the ideal I = 12Z of R. Use the Second Isomorphism Theorem to list all…

A: Certainly! The Second Isomorphism Theorem (also known as the Lattice Isomorphism Theorem) states…

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: Q5:(A) Let R=(Z12 ,+12, -12 ). Find a) Maximal ideals of R b) Is (Z12, +12) a simple group? c) Is…

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Q: Let I be an ideal of a ring R. Prove that R/I is commutative if and only if ab-ba EI for any a, b €…

A: Let I be an ideal of a ring R. We are to prove that R/I is commutative if and only if ab-ba belongs…

Q: Find all idempotent elements of the ring M_2(R).

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Q: Let R be a ring. If the only ideals of R are {0} and R itself, then R is a field.

A: We know the definition of , ideals of ring R. A non empty subset I of R is said to be an ideal of R…

Q: Is it true or false that if I is a principal ideal in a commutative ring R with multiplicative…

A: Is it true or false that if  is a principal ideal in a commutative ring  with multiplicative…

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: If A, B, C are ideals of a ring R such that BCA, prove that An (B + C) = B+(AnC) = (A B) + (AnC).

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Q: 6. Find two ideals I₁ and 12 of the ring Z such that a. I, U2 is not an ideal of Z. b. I₁ U2 is an…

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

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Q: Consider the ring R = 2Z and the ideal I = 24Z of R. Use the Second Isomorphism Theorem to list all…

A: Hi! Please refer to the attachment. Thank you!Explanation:4:

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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