Which of the following pairs R and M are not an example of a module? a. M is a subring of a ring R b. R, any commutative ring, and I an ideal of R. R=F a field; M an F-vector space d. R = Z, the ring of integers; M, any abelian group C.

Q: Let R be a ring with unity and let a∈R. Prove that if a is a zero divisor, then it is not a unit.

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Q: If S is a subring of a ring R, then S[a] is a subring of R[x]. Exercise 2.35.1 Prove this assertion!…

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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: 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: The set H= {0,2,3}is a subring of (Z6, +6,.6) integer T F ring module 6.

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Q: Let R be a ring with the left identity e. Suppose, for each nonzero element a in R, there exists b…

A: R is a ring with the left identity e.For each nonzero element a in R, there exists b in R such that…

Q: 1. Let / be a subset of the ring R. Prove that I is an ideal of R if and only if I is nonempty and x…

A: Proving that I is an ideal of R if and only if it satisfies the given conditions requires…

Q: 1. Consider the proof of the backwards direction of the theorem below. Justify each step that makes…

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Q: III. An element x of a ring R is called idempotent if x² = x. Determine all the idempotent elements…

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Q: 1. Prove that if S is a subring of a ring R then the following are true: (a) Os = OR (b) if 1R € S,…

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Q: If R is a ring and S is a subring of R, then S is  an R-module (module over R).

A: R is ring and S is a subring of R  Claim : S is an R -module (module over R)