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, be R:a.b EI = a €I and b € I (b) va, be R:a. beI a €l and be1 (e) va, be R:a.beIa el or bEl (d) No Choice

Q: 4. The ring (Z, +,.) the ideal (12) containing in the following maximal ideal ... (a) (4) (b) (5)…

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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: Let R be a commutative ring with unity, I an arbitrary ideal in R and P1,P2,...,Pn prime ideals in…

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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: Q2) Let (M₂ (R), +..) be a ring. Prove H = {(a) la, b, c €. = {(ab)la, b, c € R}is a subring of (M₂…

A: Subring Test : A non empty subset S of a ring R is a subring if S is closed under subtraction and…

Q: Suppose that Φ: R --> S is a ring homomorphism and that the imageof Φ is not {0}. If R has a…

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Q: 3. Prove that an ideal I in a ring R is the whole ring if and only if 1 e I.

A: Question: Prove that an ideal I in a ring R is the whole ring if and only if 1∈I. Proof: We have to…

Q: If ne Z not prime, then the ring (Zn, +ni'n) is a) Integral domain b) field c) Division ring d) not…

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

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

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

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

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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…

A:

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…

A:

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

A: 5.R,+,.is an infinite commutative ring and it has no nontrivial ideals then R forms 6.suppose that…

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.…

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

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Q: Suppose that R is a commutative ring with unity 1. Then if ab is a zero divisor then a or b is a…

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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…

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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: IN denotes the set of noninvertible ele conditions are equivalent: (N,+,) is an ideal of (R,+, ), p)…

A: Given N denote the set of non-invertible  elements of R.

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

A:

Q: 35. Show that the first ring is not isomorphic to the second. (a) Eand Z © Z × Zu and Z () ZXZ, and…

A: The objective is to show that the first ring of the following is not isomorphic to the second:

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

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

A: By supposinɡ S1 and S2 as two subrinɡs of rinɡ (R, +, .) To prove that S1∪ S2 is subrinɡ of R if and…

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

A:

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

A: I jave used the definition of subring

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: 1- Let (R,+,-) be aring which has property thut a=a, Ua ER.prove thatR is Commutabive ring (Every…

A:

Q: 10. Suppose that (R, +,.) be a commutative ring with identity, then R/ rad R is ... (a) semi-simple…

A: C will be right answer.

Q: Suppose R is a commutative ring with 1R# 0R. Show that if f (x) = ao + a1a + a2a ++a,n" is a unit in…

A:

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!!!

Q: The ring Z,3 has exactly------------maximal ideals 3 4 1 O O O O

A:

Q: 4) In a ring R; The sum of two non-trivial idempotent elements is not always an if we take + = is…

A: 4) We need to show that , sum of two non trivial idempotent elements is not always idempotent. We…

Q: Show that if S is a ring and M is a free S-module on a countably infinite set e1, e2, .. ., then M…

A: It is given that

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.