Let R be a commutative ring with identity. Is x an irreducible element of R[x]? Either prove that it is or give a counterexample. If the answer is no, then give a condition on R that would assure that is irreducible in R[x].

Q: Let R be a ring and x be an indeterminate. Prove that the rings R[x]and R[x2] are ring-isomorphic.

A: Given That : Two rings R[x]  and R[x2]  To Prove: Two ringsR[x],  R[x2]  are ring-isomorphic

Q: Let R be the set of all continuous functions. It is known that (R,+,×) is a ring where + and × are…

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Q: 4x-y<-1

A: Given  4x-y&lt;1 Can be rewritten as y&gt;4x-1 So we take the area above the line y = 4x -1 And all…

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: Let (R, <) be an ordered ring, and let x, y € R be positive. Prove that x < y if and only if x² <…

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Q: Let R be a ring such that a² = a for all a E R, and assume that R has an identity. Show that the…

A: Given R be a ring with unity 1 such that a2=a, for all a∈R. Let b∈R be a unit in R. Therefore b-1…

Q: Let a be an element of a ring R such that a3=1R. Prove: for any integer n, either (an)n=1R or…

A: Let a be an element of a ring R such that a3=1R.  We will find, for any integer n, either (an)n is,…

Q: 3. Assume that d € Z and √d & Q. Prove that if € Z[√d] has the property that |N₁(π)| is a prime…

A: We know that if Now given,Where p is prime number.Suppose  is not prime thenWhere are non units…

Q: Figure out what the ring Z⁄⟨−4⟩ is. 2- Find a ring that is an integral domain, but not a UFD, that…

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Q: Let R and S be commutative rings. Suppose ϕ : R → S is a ring isomorphism. A student says, “By…

A: Let R and S be commutative rings.

Q: Let R be a ring. If r3=r for all r in R, prove that R is communicative.

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