Let R be a commutative ring with an identity 1R and let J be a proper ideal with the property that every element of R that is not in J is a unit of R . Prove that J is a maximal ideal of R .

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: 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: 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: Let R be a commutative ring with identity such that every submodule of every free module is free.…

A: Let R be a commutative ring with identity such that every submodule of every free module is free. We…

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: et R be a ring and I be an ideal of R. Let p: rs-sr is in A where r,s is an element of R q:…

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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: Integral domain and prime ideal.

Q: maximal

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Q: Given that S be a proper ideal of a ring R. Then our claim is that S is contained in a maximal ideal…

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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 if and…

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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: 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: 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: Let I be an ideal of ring R. If P is a prime ideal of the quotient ring R/I, prove there exists a…

A: Let π : R → R/I be the natural ring homomorphism. Since P is a prime ideal of R/I, we have…

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: 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: Let R be a ring. For a subset X C R, let (X) denote the ideal generated by X, i.e., the smallest…

A: I have given detailed solution below thanks Explanation:Step 1: Step 2: Step 3: Step 4:

Q: Suppose M is an ideal of a commutative ring R with unity and let a ∈ R \ M . Prove that J = { m + ra…

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Q: Label each of the following statements as either true or false.   Every ideal of a ring R is a…

A: Every ideal of a ring R is a subring of R