7. Suppose that (R,+,.) be a commutative ring with identity and (I,+,.) be an ideal of R. If I is not prime ideal then (a) 3a, b e R:a. b E 1 = a ¢ I and b ¢ 1 (b) Va, b e R: a. bEI = a ¢ I and b ¢ 1 (c) Va, b e R: a.b € I = a € I or b E I (d) No Choice (a) (b) C (c) C (d) C

Q: In commutative ring with identity each proper ideal is contained in prime ideal. O T OF

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Q: 3. Let R. be a commutative ring with ideals L and M. Give an example of when LMLn M.

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Q: D. Given the set F= {0, a, b, c, d} with operations + and * and the following tables, assume that…

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Q: Q:- Consider a ving R={o,a,b,c,d, e,f.g} defired by folowing the cayley's tables. elf ef +) a e a d…

A: Solution: The given ring is R=0,a,b,c,d,e,f,g We have to calculate the characteristic of the ring R.

Q: 6. Suppose that (M,+,.) be a maximal ideal of the commutative ring with identity (R, +,.) and x E M,…

A: Since, given that (M, + .) Is a maximal ideal of the commutative ring with identity (R, +, .)  and…

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: Theorem 79. Let a, b, c, d ∈ R where R is a ring then (a + b)(c + d) = ac + ad + bc + bd. Note.…

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Q: which ring is Z_24/(3) isomorphic to? a. Z_2 b. Z_3 c. Z_4 d. Z_8 e. none of thr above

A: The given ring R is ℤ_24 = {0,1,2,...,22,23}, and the ideal is (3) = {0,3,6,...,21}

Q: Let (R, +,.) be any ring. Define R × Z = {(r,n):r E R ,n E Z} and %3D (r, n) + (s, m) = (r + s,m +…

A: In the Structure T=ℝ×ℤ additional multiplication is defined as…

Q: 7. Suppose that (R, +..) be a commutative ring with identity and (I, +,.) be an ideal of R. If I is…

A:

Q: 5. Let R be a ring and I, J ideals of R. Is IUJ always an ideal of R? Either prove that this is the…

A: The objective of the question is to determine whether the union of two ideals I and J of a ring R is…

Q: 4. a) What are the units of Z3? b) What are the units of Z4? c) Find the units in Z3 × Z4. d) Look…

A: Step 1: Step 2: Step 3: Step 4:

Q: 1) Let R be a commutative ring and P,Q ideals in R. The product of the ideals P, Q is defined as т…

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Q: 3. In a Ring (R, +, *) let '0' be identity of '+' and '1' be identity of only one element. unique…

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Q: Hom worke: Consider the ring (Z [√3], +,.), Let A={ a+b√3:a, be Z₂} Is A subring of Z[√3]?

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Q: 9. Let 11 and 12 be ideals of the ring R. Prove that the set = {a,b,+ a2b₂+ is an ideal of R. +abla,…

A: To prove that I1​I2​  is an ideal of the ring  R , we need to show that it satisfies the properties…

Q: 6. Let I and J be ideals in the ring R. (a) Prove that InJ is an ideal in R. (b) Prove that I+J =…

A: Since you have posted a question with multiple sub-parts, we will solve first three subparts for…

Q: 7. Aring with identity (R. +..) is commutative if: (a) Ris an integral domain () (a+ b) - a + Zab +…

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Q: . If A, B and Ç are ideals of a ring R, prove that A (B+ C) = AB+AC.

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Q: '. Suppose that (R, +,.) be a commutative ring with identity and (I, +,.) be an ideal of R. If I is…

A: R is ring with unity and I is ideal of R.

Q: 6. Suppose that (M,+,.) be a maximal ideal of the commutative ring with identity (R, +,.) and x E M,…

A:

Q: Let R be a ring. Let p= ax + b and q = cx² + dx + e be elements of R[x], for a, b, c, d, e E R. True…

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Q: Theorem 3. Let R be a ring. Then for all a,b,c E R, 1. a0 =0= 0a. 2. a(-b) = -(ab) = (-a)b. 3.…

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Q: Let I,J be the ideals of a ring P. Show that the sets (a)lJ = {a,b, +.….+a,b, :a, E1,b, E…

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Q: Suppose A and B are ideal of the ring R. if A C

A: I have used definition of ideal of ring.