Let R be a commutative ring with unity. Fix two elements a, b ∈ R. Prove that if a = bt for some t ∈ R, then ⟨a⟩ ⊆ ⟨b⟩.   Let R be a commutative ring with unity. Prove that ⟨1R⟩ = R

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A: Let R be a Euclidean domain with measure d. Also given that a and b are associates in R. Then from…

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A: we will use sub ring test.

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A: Let R be a ring with unity 1R and let S be a subring of R containing 1R. If r∈R is a unit of R and…

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A: Solve the following

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Q: If in a ring R every x E R satisfies x2 = x, Prove that R must be commutative.

A: Answer and explanation is given below...

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A: Yes , it is possible for R[x] to be a PID ( assuming R[x] is PID ) without being a Euclidean domain.

Q: If R is a commutative ring with unity and a e R, then (a) = {ar:reR}=aR. %3D %3D

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A: Since you have asked multiple questions so as per guidelines we will solve the first question for…

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Q: et R be a ring and assume a∈R is not a zero divisor. If ab=ac for some b,c∈R, then b=c

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Q: Show if R is a commutative ring then R[x] is also a commutative ring.

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Q: If R1 and R2 are subrings of the ring R, prove that R1 n R2 is a subring of R.

A: R1 and R2 are subrings of the ring R, prove that R1∩R2 is a subring of R

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A: R is ring with unity and I is ideal of R.

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: Let R be a commutative ring with 1 ≠ 0. Prove that R is a field if and only if 0 is a maximal ideal.

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