If N is an ideal of a ring R, which of the following statements is NOT always true? N is a subring of R. m – n eN for all m, n E N. O Option 1 Option 2 Na CN for all a E R. Na = aN for all a e R. Option 3 Option 4

Q: IF J is nil left ideal in an Artinian ring R, then J is

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Q: For any ideal A in the ring R, prove that A+A=A

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Q: Let R be a commutative ring with identity and P be an ideal. Prove that if R/P is a ring that has…

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Q: Show that the intersection of any collection of subrings of a ring R is also a subring.

A: Sol:- To show that the intersection of any collection of subrings of a ring R is also a subring, we…

Q: Give an example of a ring homomorphism from a commutative ringR to a ring S that maps a zero-divisor…

A: Given: Commutative ring R and S.We need to provide an example of a ring homomorphism from R→S that…

Q: 1. Prove: The square of any integer n is of the form 3k or of the form 3k+1 for some integer k. 2.…

A: Hey, since there are multiple questions posted, we will answer first question. If you want any…

Q: IfE isa ring of even inte gers, then show that an ideal (4), generated by 4 is a maximal ideal of E.

A: If E is a ring of even integers, then our claim that an ideal (4), generated by 4 is a  maximal…

Q: of Let {S; : i ¤ A} be a non - empty family of prime ideals a ring R and suppose ¹f i, j ¤ A, then…

A: Let be a non-empty family of prime ideals of a ring R and suppose if i,j∈?, then we will show that…

Q: Let R be a commutative ring. If I and P are ideals of R with P prime such that I ¢ P, prove that the…

A: The ideal quotient of P and I is P:I=x∈R : xI⊂P which is again an ideal of R. Given that P is a…

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: If A, B, C are ideals of a ring R such that BCA, prove that An (B + C) = B+(AnC) = (A B) + (AnC).

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Q: If E is a ring of even integers , then show that an ideal  (4) , generateed by 4 is a maximal ideal…

A: Let S=(4) and T be an ideal of E such that S⊆T.Then there exists an element a∈T such that a∉(4),that…

Q: Let {S;:i E A} be a non-empty family of prime ideals of a ring R and suppose if i, je A, then either…

A: Let be a non-empty family of prime ideals of a ring R and suppose if i,j∈?, then we will show that…

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: Show that the union of a chain I, CI,c of ideals of a ring R is an ideal of R. (This exercise is…

A: Given: We are given a chain of ideals in a ring Objective: The objective is to show that the union…

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: Either prove or give a counterexample: If R is a ring with zero divisors, then R[x] is a ring with…

A: (A) Given that R is a ring with zero divisors.

Q: Prove that the triple Z. , 0 has no zero divisors for prime p E P.

A: Note that : Since you have posted multiple questions, we will provide the solution only to the first…

Q: If S be an ideal ofg a ring Z of all integers. Then S is maximal if and only if it is generated by…

A: If S be an ideal ofg a ring Z of all integers. Then S is maximal if  and only if it is generated by…

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: If S be an ideal ofg a ring Z of all integers. Then S is maximal if and only if it is generated by…

A: is an ideal of ring . We have to show if  is maximal if and only if it is generated by some prime…

Q: unity. Show, by example, that a maximal ideal in R need not be a maximal id

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

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Q: Q[V5 is Principal Ideal Domain. True O False

A: Since you have asked multiple questions, as per our company guidelines, we will solve first question…

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…