Let ? be a commutative ring with 1 and ? be a proper ideal of ?. Prove that ? is prime if  and only if ?/? is an integral domain.

Q: Let R be a ring with unity and let a∈R. Prove that if a is a zero divisor, then it is not a unit.

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Q: Let R be a ring and let I be an ideal of R. Prove that the factor ring R/I is commutative iff rs-sr…

A: We have to prove that factor ring R/I is commutative iff rs-sr is in R for all r and s in R.

Q: if A and B are ideals in a ring R such that A intersect B ={0}, prove that for every a in A and b in…

A: Let A and B are ideals of a ring R such that A∩B=0

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…

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Q: As a ring, z is isormorphic to nZ for all n>1. True False

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Q: Let R be a commutative ring with 10. Prove that R is a field if and only if 0 is a maximal ideal.

A: If R is a field, then prove that {0} is a maximal ideal. Suppose that R is a field and let I be a…

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: Let R be a commutative ring. Prove that HomR(R, M) and M are isomorphic R-modules

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Q: Let R be a ring with a subring S. Prove or disprove: If a ∈ R is nilpotent and a ∈ S, then a is also…

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Q: Let R be a commutative ring with unity and let M be a maximal ideal of R such that M2 = {0}. Show…

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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 R is a commutative ring with unity, show that every maximal ideal of R is a prime ideal.

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Q: Suppose that R is a commutative ring and |R| = 30. If I is an idealof R and |I| = 10, prove that I…

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Q: Let R be a finite commutative ring with identity. Then every prime ideal of R is maximal True O…

A: To prove that every prime ideal of R is maximal.

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: a is a unit in a ring R with unity, then a is not a zero divisor

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Q: Let I be an ideal of a commutative ring (R,+.)with unity, then I is no prime ideal iff RM is…

A: We know characterstic of an integral domain is either prime or zero. Let I be an ideal of a…

Q: Is it true that if S is a unital subring of a unital ring, then the identity elements of the two…

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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: For any element a in a ring R, define (a) to be the smallest ideal of R that contains a. If R is a…

A: An ideal is a non-empty sub set I of a ring R, such that

Q: Let R be a commutative ring with identity. Using the homomorphism theorem (Theorem 16.45) and…

A: Recall that in a ring A  not necessarily commutative and with an identity, an ideal M⊂A is a maximal…

Q: Let I be the set of all elements of a ring R that have finite additive order. Prove that I is an…

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Q: Suppose R is a commutative ring and |R|= 30. If I is an ideal of R and |I| = 10, prove that I is a…

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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: Label each of the following statements as either true or false.   The only ideal of a ring R that…

A: Given Statement, The only ideal of a ring R that properly contains a maximal ideal is the ideal R.…

Q: Let R be a ring with unity and assume a ∈ R is a unit. Prove that a is not nilpotent.

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Q: 2. Let R be a commutative ring with unity. If I is a prime ideal of R, prove that I [x] is a prime…

A: Given R be a commutative ring with unity and if I is a prime ideal of R. Then we have to prove that…

Q: If I is an ideal of a ring R, prove that I[x] is an ideal of R[x].

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Q: Show that a commutative ring R with unity is a field if and only if its only two ideals are (0) and…

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Q: Let R be a commutative ring with unity and let M be a maximal ideal of R such that M2 = {0}. Show…

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Q: If A and B are ideals in a ring R such that A n B = {0}, prove that for every a E A and b E B, ab =…

A: Explanation of the answer is as follows

Q: Let R be a ring and S a subring of R. Prove that if 1R ∈S, then S has unity and 1S =1R

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Q: Let R be a commutative ring with unity. If I is a prime ideal of R prove that I[x] is a prime ideal…

A: Let R be a commutative ring with unity. If I is a prime ideal of R we have to prove that I[x] is a…

Q: Let R be a commutative ring with unity and let M be a maximal ideal of R such that M2 = {0}. Show…

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Q: 1. Let R be a commutative ring with unity and let a e Rbe fixed. Prove that the subset Ia = {x E R:…

A: i have provided the detailed proof in next step

Q: Let I= {(a, 0)| a eZ}. Show that I is a prime ideal, but not a maximal ideal of the ring Z×Z. Id if…

A: We knew that an ideal I of a ring R is said to be prime ideal if for a ,b ∈ R and ab ∈ I this imply…

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: 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: Let M be a proper ideal in a Boolean ring R with unity. Prove that (1) R/M is a Boolean ring and…

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Q: Let R be a commutative ring with unity and let N={ aER | a"=0 for nez*, n>1}. Show that N is an…

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Q: 4: prove that Let R be a commutative ring with identity and I be a maximal ideal of R. Then the…

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Q: The ring Zpg?, has exactly-------------maximal ideals O 2

A: 3

Q: Let I be a maximal proper ideal of commutative ring with identity R. Prove that R/I is a field.

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

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Q: Let R be a commutative ring. Prove that HomR (R, M) and M are isomorphic R-modules.

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Prove that if (I,+,.) is an ideal of the Ring (R,+,.) then rad I= In rad R ???

A: Solution :

Q: Let R be a commutative ring with 1 ≠ 0. Prove that R is a field if and only if 0 is a maximal ideal.

A: We are given that R be a commutative ring with unity. We have to show that  R is a field if and only…

Q: 5. Let A and B be two ideals of a commutative ring R with unity such that A +B = R. Show, that AB=A…

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