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: The ring Z pg?, has exactly------------maximal ideals 2 3 1 4

A: An ideal I in Zn  is maximal if and only if I=⟨p⟩ where p is a prime dividing n.

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

A:

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

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A:

Q: (4) The ring (Z, +,.) has the following not maximal ideal (a) ((11), +,.) (b) ((31), +, .) (c) ((0),…

A: The given question is solved with explanation below.

Q: 4) Define on the cartesian product of sets R:=R x R = {(^, z) | A, z E R } an addition and a…

A: We first show that R is a commutative ring. That is we show that R is commutative group under…

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Q: Q:Let S and Szare two subrings of a ring (R, +..), prove that S, US2 subring of R iff either S, C S…

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Q: (3) Let A be commutative ring with identity, then A has just trivial ideals iff A is ....... O…

A: Here you have posted multiple question, So as per the policy I can answer only first question for…

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

A:

Q: Let (R, +, ) be a ring, a, b e R with a + 0. Then the equation ax = b has a solution in R.

A:

Q: 9. Suppose that (R,+, .) be a commutative ring with identity and x E rad R, then ..... .... (a) (x)…

A:

Q: Let R is ring of real numbers and *, (2) O defind on R^2 as follows, (a, b), (c, d)ER^2 then (a, b)*…

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Q: 3) Given a commutative ring with unity 1 in R; where R is a ring with two maximal ideals M₁ and Show…

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Q: 1- Let ý:R, » R, be a ring homomorphism such that Kerø =. Then, o is a) 1-1 b) onto c) Both 1-1 and…

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Q: Let U = {a, b}. Define addition and multiplication in P(U) by C +D = CU D and CD = Cn D. Decide…

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Q: Show that the set 4Z U 5Z is not a subring of the integer ring Z:

A: Z = { ....., -3, -2, -1, 0, 1, 2, 3, ..... }. 4Z = { ...., -8, -4, 0, 4, 8, .... }. 5Z = { ....,…

Q: 4 If R is a commutativ ring, show that the characteristic of R[X] is the same the characteristic of…

A: Given

Q: 9. Suppose that (R,+, .) be a commutative ring with identity and xE rad R, then (a) (x) = R (b) 1-x…

A:

Q: 5. An element x in a ring R is called idempotent if a2 = x. Prove that if a is an idempotent element…

A:

Q: If A and B are ideals of a commutative ring R, define the sum of A and B as: A+B = {a+b| a € A, b €…

A: Let A and B be two ideals of a ring R. If  A and B are ideals of a commutative ring of R with unity…

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: 1- Let ý:R, » R, be a ring homomorphism such that Kerø = . Then, ø is a) 1-1 b) onto c) Both 1-I and…

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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: 5. Suppose that (R, +,.) is an infinite commutative ring and it has no nontrivial ideals, then R…

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Q: 24. Let (R, +,) be a commutative ring with identity and a ER be an idempotent which is different…

A: R, +,· is said to be commutative ring if Suppose R is a non empty set such that for any two elements…

Q: The set of all zero divisors of the ring Z6 is اختر احدى الاجابات O {2, 3, 4} O (1, 3, 5) O {1, 2,…

A: Z6={0,1,2,3,4,5}Since 2 · 3 ≡6≡ 0 (mod 6) and 3 · 4 ≡12≡ 0(mod 6)However, 1 and 5 are not zero…

Q: 1. Show that x^2 - y^2 = (x – y)(x + y) for all x, y in a ring R if and only if R is commutative.…

A:

Q: Let R, , O is a ring under two composion e and O üs follows ü e i; = a + b + 1 and aOb = ab + a + b…

A:

Q: 1. Suppose that (R,+,.) is a ring and I is not maximal ideal in R. Then .... ..... (а) 1 3D R (b) 3J…

A:

Q: (4) The ring (Z, +,.) has the following not maximal ideal (a) ((11), +,.) (b) ((31), +,.) (c) ((0),…

A: Let R be a ring. A two-sided ideal I of R is called maximal if I ≠ R and no proper ideal of R…

Q: Q1: Let S, and Szare two subrings of a ring (R, +,.), prove that S, USz is subring of R iff either…

A:

Q: (a) Let R be a commutative ring with M being maximal ideal in R then R/M is a field. (b) If R is a…

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Q: lisprove: that the triple <Z,,, constitutes a ring. et of units U(Z₂), and state whether (U(Z₁), Ⓒ)…

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Q: An cxample of an infinite non-commutative ring with identity is: M,(Z) Ma (2z10) O This Option O…

A:

Q: 1. Suppose that (R, +,.) is a ring and I is not maximal ideal in R. Then .. (a) I = R (b) 3/ ideal…

A:

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

A: Prime Ideal:- An ideal P of a commutative ring R is prime if it has the following two properties:…

Q: 9. Suppose that (R,+,.) be a commutative ring with identity and x E rad R, then (a) (x) = R (b) 1- x…

A:

Q: 1- Every Boolcan algebra (B, V, A), becomes a Boolean ring (B, +, .) on defining addition and…

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Q: Consider the ring (Q,Ð,0), where the "addition" O and "multiplication" ® are defined as follows:…

A:

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

A:

Q: The center of a ring R is {z ∈ R : zr = rz for all r ∈ R}, i.e. the set of all elements which…

A:

Q: If R be a commutative ring with identity and a ∈ R is idempotent different from 0 or 1. Prove that R…

A: Given that R is a commutative ring  with unity. Also, a∈R  is an idempotent element ⇒a2=a ( a≠0,1)…

Q: The set of all zero divisors of the ring Z6 is اختر احدى الدجابات O (2,3,4) O (1,3,5) O (1,2,3,4,5)…

A: The set of all zero divisors of the ring Z6 is (0,2,4)

Q: Give the following theorem (without proof): If (R, +, ·) is a ring, and S C R then what is the…

A: That's easy. Thumb up. Have a great day!!!

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A: It is given that

Q: Show that a ring R is commutative if and only it a - b = (a+ b) (a - b) for all a, be R.

A: Proof. Let R be commutative. Then ab = ba for all a,b ∈ R.

Q: 17. Given that (I, 1.) in an ideal of the ring (R,+,), show that a) whenever (R,1,) is commutative…

A: Definition Two-sided Ideal of a ring : An ideal I of a ring R is called a two-sided ideal if it…

Q: Theorem 2.7. IfI, I, I2, . .., I, are ideals in a commutative ring R, then: (i) Rad (Rad I) Rad I;…

A: The solution for the above question is as shown below.