Let R be a commutative ring with unity and let M be a maximal ideal of R such that M2 = {0}. Show that if N is any other maximal ideal of R, then N= M.

Q: Find all maximal ideals of Z_12:

A: Given ring is Z_12

Q: Let R be a commutative ring with unity and let S ‡ R be an ideal of R. Then R/S is an integral…

A: is a commutative ring with unity and  is an ideal of .We have to prove  is an integral domain if and…

Q: Let BCR be a subset of R. Prove that there is an ideal I of R such that B n I has at most one…

A: To show that ideals I and J  exist satisfying conditions 1) and 2)

Q: give an example of An ideal in Z that is not a prime ideal.

A: The objective is to write an example of an ideal in ℤ that is not a prime ideal.

Q: (9) Every primary ideal is a maximal ideal. От OF

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Q: Let R be a ring with unity which has exactly one maximal left ideal M. Show that the only…

A: Given: R is a ring with unity which has exactly one maximal left ideal M. To show: The only…

Q: Prove that Let R be a commutative ring with unity. Then every maximal ideal of R is a prime deal.

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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: 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: maximal

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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: Prove that every non-zero prime ideal of Z is a maximal ideal.

A:

Q: Let R = {(a,, a,, a, ...)}, where each a; E Z. Let I = {(a,, a, az, ...)}, where only a finite…

A: Here given that R=a1, a2, a3......., where each ai∈Z. Consider, I=a1, a2, a3........ It has to be…

Q: (B) Find all maximal ideals of (Z8, +8,8).

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Q: Let R be a ring and let I a minimal left ideal  of R. If M is a faithful simple left R-mod

A: R be a ring and I be a minimal left ideal of R

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: a) prove that the set J of all polynomials in z whose constant terms are divisible by 3 is an ideal…

A: (a). To prove that the set J of all polynomials in z whose constant terms       are divisible by 3…

Q: 2. Use the Fundamental Homomorphism Theorem for rings (FHT) to prove (the ideal generated by x-4)…

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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: 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: Let Rbe a commutative ring with unity and let S‡ Rbe an ideal of R. Then RS is an integral domain if…

A:

Q: Let P be a prime number. Let Zp) be the following subset of Q: a Z(p) = {₁ where a € Z, bЄ N, such…

A: Let p be a prime number and Zp be the following subset of Q. Zp=ab, where a∈Z , b∈N , such that p…

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

A: It is given that, M is a proper ideal in a Boolean ring R with unity.

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

A: Given below the detailed solution

Q: b\ Find the domain of the functions: (1) y = (2) y = = (3)f(x) =

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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: Let R be a commutative ring with an identity 1R and let J be a proper ideal with the property that…

A: Given R is a commutative ring with identity 1R                   J is a proper ideal of R R/J…

Q: Let R be a commutative ring with unity. Then every maximal ideal of R is a prime deal.

A: Let R be a commutative ring with unity. Then we have to show that every maximal ideal of R is a…