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 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 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: 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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