3) Let R be a commutative ring with 1. Let A and B be two distinct maximal ideals of R. Show that AB = An B.

Q: If Z is integers ring, then each non zero element in a proper ideal of Z is......... O (i)…

A:

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

A: Given that (R, +, .) be a commutative ring with identity and (I, +, .)  be an ideal of R such that I…

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: Let R be a ring with a finite number n of elements. Show that the characteristic of R divides n.

A: Given :  R is a ring with n elements. To prove :  The characteristic of a ring R divides the number…

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…

Q: Suppose that R and S are commutative rings with unities. Let ø be a ring homomorphism from R onto S…

A:

Q: Verify in the following statement involving the ideal generated by (a1, a2 , ... , ak ) in the ring…

A: To show that 2,3=ℤ Let 2,3=2x+3y: x,y∈ℤ -------(1) gcd2,3=1 , Thus 2m+3n=1 for some…

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: Verify in the following statement involving the ideal generated by (a1, a2, ... , ak )) in the ring…

A: To prove that: (2, 5)=ℤ

Q: Suppose A is a Noetherian unital commutative ring and I is an ideal of A. Prove that A/I is…

A: Some basic results of module theory required to understand the following proof better . If R is a…

Q: Let R be a ring with unity. Show that {а)3D( 2 хау: х, у є R}. finite

A:

Q: Prove that a ring with unity R has a unique maximal left ideal M if and only if R\M is the set of…

A: Prove that a ring with unity R has a unique maximal left ideal M if and only if R\M is the set of…

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

A:

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

A:

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…

A:

Q: 16. If R is a field, show that the only two ideals of R are {0} and R itself.

A:

Q: Show that if R is a ring with unity and N is an ideal of R such that N R, then R/N is a ring with…

A: Since we have given that R is a ring with unity and N is a proper  ideal of R . We prove R/N is a…

Q: Show that the nilpotent elements of a commutative ring form a subring

A: We have to prove that the nilpotent elements of a commutative ring form a subring. In order to do…

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

A:

Q: 21. A ring is said to be a local ring if it has a unique maximal ideal. If (R,+,) is a local ring…

A:

Q: 20. Suppose R is a commutative ring and [R]= 30. If I is an ideal of R and |I| = 10, prove that I is…

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: 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: Prove that in a ring R having exactly one maximal ideal M, the only idempotents are 0 and 1.

A: Assume R is local and let M=R∖Rx. By assumption M is an ideal. It is also maximal because any ideal…

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

A:

Q: Show that the ideal of (5) in the ring of integers Z is the maximal ideal.

A: An ideal A in a ring R is called maximal if A ≠ R and the only ideal strictly containing A is R. In…

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: Let K be integer ring module 12 and let I=([4]) and J-([6]) be ideals of K. Then ([0])

A: Let * be integer ring module 12.  Let  I=4J=6

Q: Let R be a finite commutative ring with unity.  Show that every prime ideal in R is a maximal ideal.

A: We have to show that if R be a finite commutative ring with unity then every prime ideal in R is a…

Q: If A and B are ideals of a commutative ring R with unity and A + B= R, show that A N B = AB.

A: Given Data: A and B are ideal of a commutative ring R with unity And A + B = R   For A⋂B = AB, The…

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

A:

Q: If R is a commutative ring with unity and A is a proper ideal of R, show that R/A is a commutative…

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: The cancellation laws for multiplication are satisfied in a ring R, if R has zero divisor.

A:

Q: Exhibit a commutative ring R and an element x E R such that Z CR and x is NOT prime but irreducible…

A: Take R = Z[i√5]  Clearly, R is commutative ring and Z ⊆ R  Also, 2,3 ∈ R are not prime but…

Q: Given a commutative ring R considered as a module over itself, determine all the module…

A: Given: A commutative ring R is considered as a module over itself.   To determine: All the module…

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

A:

Q: 8. List all ideals of the ring Z12-

A: Ideals of the ring Z12

Q: Let M be a commutative ring with identity x and R be a maximal ideal of M. Then prove that R is…

A: Prime Ideal : A prime ideal S of a commutative ring R is a proper ideal of R such that a, b∈R and…

Q: elements

A:

Q: Prove that a finite ring R with unity and no zero divisors is a division ring.

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Q: Suppose I, J be ideals of a commutative ring R. Prove that IJ cIn).

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: Suppose that R is a commutative ring with unity and that I is an ideal of R. Prove that the set of…

A: Given: R is a commutative ring with unity and that I is an ideal of R. To prove: The set of all x∈R…

Q: Let R be a ring and M be an R-module. Ther

A:

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: is union of two ideal rings R, an idea of R? prove or give counter example

A:

Q: 21. A ring is said to be a local ring if it has a unique maximal ideal. If (R,+, ) is a local ring…

A:

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…

A: