3. a) Write the definition of a characteristic of a ring. b) Find the characteristic of the ring (Z/20Z)/ (5Z/20Z).

Q: Let I be an ideal in a ring R with unity. Show that I = R if and only if I contains a unit.

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Q: 3. Let R be a commutative ring with unity. The intersection of all maximal ideals of R is called the…

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

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Q: Let be a commutative ring with unity of characteristic 3. Compute and simplify (a + b)° for all…

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Q: 12. Let (I,+,') be an ideal of the ring (R,+, ·). Prove that (I,+, ·) is a primary ideal if and only…

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Q: Find all ideals of Z3 [i] . Show your work. (-

A: ℤ3i is clearly a field  because ℤpi is a field if p≡3 (mod 4). Thus 3≡3(mod 4) therefore ℤ3i is a…

Q: Consider the ring ℤ12. List the elements of the following principal ideals: (i) ⟨4⟩  (ii) ⟨9⟩…

A: Given ring is ℤ12.

Q: Let R be a commutative ring that does not have a unity. For a fixed a e R prove that the set (a) =…

A: Let R be a commutative ring that does not have a unity. For a fixeda∈ℝ, we need to prove that :…

Q: It is known that (ℤ,⋆,⊙) where ? ⋆ ? = ? + ? − 1 and ? ⊙ ? = ? + ? − ?? is a ring. Does it have…

A: Please check step 2 for the solution.

Q: (b) Show that any nonzero element of the ring QIV2 = {a + bv2 | a, b e Q} is invertible, that is,…

A: b)  We have given that ,  ℚ2 = a + b2 / a , b ∈ ℚ We need to show that , for any non-zero element of…

Q: is the ring 2Z isomorphic to the ring 4Z??

A: We can apply definition

Q: Show that the centre of a ring R is a sub ring of R. And also show that the centre of a division…

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Q: If A and B are ideals of a ring, show that the product of A and B, AB = {a,b, + a,b, +· · · + a,b,l…

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Q: Let R be a ring with 1 0. Prove or disprove: (a) if R has no ideals other than {0} and R, then R is…

A: Given statement is false. Justification is in step 2

Q: The cancellation laws for multiplication are satisfied in a ring T F R, if R has zero divisor.

A: Here, given that The ring R with the cancellation law for multiplication holds in R. Let a,b,c∈R if…

Q: (b): Define reducible and irreducible elements in ring R and find all the reducible and irreducible…

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Q: 21. A ring is said to be a local ring if it has a unique maximal ideal. If (R,+,) is a local ring…

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Q: 21. If R is a ring, prove that R[x] ~R, is the ideal generated by x.

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Q: 18. Let (R, ,) be a commutative ring with identity and let N denote the set of nilpotent clements of…

A: a). As given that N is nonempty as 0∈N.  an=0 (n= positive integer) Then r∈R ran=rnan=0 Let take a,…

Q: The ring Z75 has exactly-----------maximal ideals 4 O 2 6 3

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Q: Let R be a ring with m elements. Show that the characteristic of Rdivides m.

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Q: Let a and b be elements of a ring R. Prove that the equation a+ x= b has a unique solution.

A: a and b are elements of ring R. We have to show that (a+x)=b has a unique solution.

Q: 7. The number of distinct homomorphisms from the ring (Z, +,.) onto the ring (Ze, +,.) is (a) 0 (b)…

A: Number of distinct onto homomorphism is

Q: Consider the ring of polynomials Q(z) , x²-1∈Q(z) Is aprinciple ideal ? Is a maximal ideal?

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Q: 17. Let H and K be ideals of a ring R. Show that HNK is an ideal of R.

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Q: Let R be a ring with unity 1. If 1 has infinite order under addition then the characteristic of R is…

A: The order of an element a in a ring R is the least positive integer n such that n·a=0. If no such…

Q: b) Prove that, if S is a ring with characteristic 0, then S infinite.

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Q: given an example for a ring R has ideal I such that : VI #I + IC VI

A: Note:- If you don't understand anything or want anything else in this problem then please resubmit…

Q: Prove that a nonzero commutative ring with unity R is a field if and only if it has two ideals (0)…

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Q: 1. Let M and N be ideals of a ring R and let H = {m+n|m E M, n E N}. (a) Show that H is an ideal of…

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Q: 30. Let R be a ring with identity lr and S a subring of R with identity 1s. Prove or disprove that…

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Q: 7. a) Prove that every field is a principal ideal ring. b) Consider the set of numbers R {a+ bV2|a,…

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Q: Let R be a commutative ring of characteristic 2. Prove that : (a+ b) = a² +b² = (a - b)? v a, be R.…

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Q: Theorem integrl The Ring Zp H is an Domarn Pis prime-

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Q: 7. a) Prove that every field is a principal ideal ring. b) Consider the set of numbers R = {a+…

A: a) Let F be a field. We know that field has no proper ideals. The ideals of F are 0 and F only. The…

Q: Q2: Prove that the intersection of any two ideals of a ring R is also an ideal.

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Q: If R is a commutative ring with unity, show that the characteristic of R[x] is the same as the…

A: If R is a commutative ring with unity, show that the characteristic of R[x] is the same as the…

Q: 2- An example of two ideals A and B of the ring (Z12. +2-u) such thata B are: (a) A and B (c) A- and…

A: First we find all subgroup of Z12

Q: a) If U and V are ideals of a ring R and let UV be the set of all those elements which can be…

A: We are given that U and V are the ideals of a ring R. Now, we need to show that UV is an ideal of R.…

Q: The ring Z,3 has exactly-------------maximal ideals

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Q: If R is a commutative ring, shw that the characteristic of R[x] is the same as the characteristic of…

A: Given: R is commutative ring

Q: (B) Explain the relationship between an a) Ideal and subring. b) Prime ideal and maximal ideal. c)…

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Q: Give an example of ring elements a and b with the proerties that ab = 0 but ba does not equal 0.

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Q: 8.38. Find the characteristic of each of the following rings. 1. Z4 O Z10 2. M2(Z3)

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Q: If I1 and I2  are two ideals of the ring R, prove that Ii n 11 ∩ I 2  is an ideal of R.

A: Given  I1 and I2  are two ideals of the ring R To prove : I1∩I2 is an ideal of R.

Q: 3) Let R be a commutative ring with 1. Let A and B be two distinct maximal ideals of R. Show that AB…

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Q: Theorem(3):- ( Third Fundamental Theorem ) If (K, +,.) and (L,+,.) are two ideals in a ring (R,+,.)…

A: Given a ring (R,+,∙) and two ideals  (K,+,∙) &  (L,+,∙) where K⊆L.We have to prove  RKLK≃RL…

Q: The ring Z-[i] has no proper ideals

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Q: Find the maximal ideals and the prime ideals in the ring Z6.

A: We have to find the maximal ideals and the prime ideals in the ring Z6

Q: 21. A ring is said to be a local ring if it has a unique maximal ideal. If (R,+, ) is a local ring…

A: