1) Let R be a commutative ring and P,Q ideals in R. The product of the ideals P, Q is defined as т P.Q := {Ea; • b; | m e N, a; e P, b; e Q }, i=1 and the sum is defined as P+ Q := {a+b|a E P , be Q } . P +Q := (i) Show that P·Q, P+Q, and PNQ are ideals in R, and that P·Q C PNQ. (ii) Show that if P + Q and both are maximal ideals then P· Q = PnQ.

Q: 7. (a) Let be an increasing sequence of ideals in a ring R. Prove that the union of all these 1, is…

A: Both the sub-parts are solved below.

Q: A https://docs.google.com/forms/d/e/1FAlpQLSeUGYc-JH5nqoxlHtmLUWA46Q 150% Search Let a E R which…

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Q: (5) The ring (Z6, +6,•6 ) has that rad Z, (a) ((0), +6.-6 )

A: Given: To explain the correct option of the given statement as follows,

Q: Let M2(Z) be a ring and =(( ), a,b e Z}. Then S is ideal %D .of M2(Z) l choose one true Error

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Q: 3. In the ring (Z12, +12,-12 ), the following ideal is maximal (a) (2) (b) (3) (c) (a) and (b) (d)…

A: An ideal of any ring is maximal ideal if it is the largest ideal. Suppose P and Q are two ideals of…

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: 2. Let B and C be a ideals of a ring R. Prove that the set B - C = {teR such that t = f -g for some…

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Q: If (R,+,.) be aring with identity and A is : ideal of R, then the identity element of R/A is * O A+1…

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Q: Problem 4 Let I = (6, 15, 27) be the ideal in Z generated by 6, 15, and 27. Since every ideal in Z…

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Q: K be ideals of a non-commutative division division ring. Then prove or disprove that-I+ K is ideal…

A: (a) Given that I,J,K are ideals of non-commutative division ring. The objective is to show that…

Q: a) The ring R[x, y]/(x + 1) is a field. b) The ring Z[x]/(7) is a principal ideal domain

A: a) The ring R[x, y]/(x + 1) is not a field. To determine if it is a field, we need to check if every…

Q: For any ideal (1, 1.) of the ring (R, 1.), lefine the set V by Vi - {rER " E I for some nEZ4}. %3D…

A: Note : Providing all the proof : considering R to be a commutative ring . We will use the following…

Q: Which of the following statement is false? Ta) The ring (Q, +, .} is an integral domain. (b) The…

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Q: 2- Let f be an isomorphism from the ring (R, +,) to the ring (R', +','). If (I, +;) is an ideal of…

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

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Q: 1. Let R be the ring of all functions f: R → R, with operations of pointwise addition and…

A: Ideal:  Let  is a ring. A nonempty subset  of a ring  is call an ideal of , if following are true.…

Q: For what values of a does the homomorphism system (b – 1)x + 2y = 0 2x + (b – 1)y = 0 Have a…

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Q: Which of the followi:ng statement is false? Ta) The ring (Q, +, .} is an integral domain. (b) The…

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Q: 5. In the ring (Mat,x2(Z), +,.), if A € Matzx2(Z) then the ideal (A) = (a) {B. A: BE Matzx2(Z)} (b)…

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Q: à) In Z18, find all proper ideals: >) Find all the elements (cosets) of Zg/I, where I is the ideal…

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Q: Show that the power set of 2, 2º, is algebra on 2. an

A: A σ-algebra is a type of algebra so, every sigma algebra is an algebra.        (1)

Q: 7. If f is a homomorphism from the ring (R, -+, ) onto the ring (R', +', '), show the following: a)…

A: Given that f is a homomorphism from the ring (R,+,.) onto the ring (R',+',.') The objective is to…

Q: Prove that if (I,+,) is an ideal of the ring (R,+, ), then rad I In rad R. %3D

A: The term radical is used when we think about ideals and when we talk about ideals definitely…

Q: Show that for any k-algebras A1, A2,..., A,, we have (A1 x A2 x ...x A, )P S A" x A" x as algebras.

A: Let, A1,+,*,A2,+,*,...,Ar,+,* be the K-algebras, where + is the addition from 1 to r o which A1 is…

Q: Let (R,+,.) be acommutive ring with identity and a∈R be an idempotent which is different from 0 or 1…

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Q: 1 Properties of the Cross Product: lamat, 1.) ||lox b|| = |lo||-·||bl| - sind, where O & [alt] is…

A: As this question has multiple subparts, hence only first five subparts will be done.

Q: În a ring R, show that i) Oa = a0 = 0 ii) a(- b) = (- a)b=- (ab).

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Q: show by of R then IUI is not ideal example if In and Iz are icdeals an

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Q: Let an ideal I = {(a, 0): a ∈ Z} of the ring R = Z ⊕ Z be given. I, R is a prime Is it the ideal

A: To prove: I is an ideal and it is prime ideal of R. Here, R=Z⊕Z Let ,I=a,0:a∈ℤ A non empty subset I…

Q: Check whether i e Q(v-2) or not.

A: Let p,q ∈Q such thatp2-2q2+i22pq=p+q-22=-1then either p=0 or q=0, as22pq=0 Since -1 has no imaginary…

Q: a b r s - Let R а, b, d 0 d r, s, tE Z, s and S even}.ir If S is an ideal of R, what can you say…

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Q: 2/32 a) = Z5 52/32 쓴 Z X R Z × R, 4Z × {0} b)

A: We have to show that the following are isomorphic:   and  .   and  . First Isomorphism Theorem:  Let…

Q: Consider a ring ZI/2)={a + b[/2]|a,bEZ}where addition and multiplication on ZI/21is defined as (a +…

A: A ring is commutative if all the elements of the ring commute under the second operation. Clearly,…