Abstract Algebra: Subrings and Subfields.  Let e be an idempotent of a ring R. Prove that the set :eRe={ere |  r in R} is a subring of R with e as the identity element.

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Q: 5. Use the Forward-Difference method to approximate the solution to the following parabolic partial…

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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: Abstract Algebra: The center of a ring R is {z ∈ R : zr = rz for all r ∈ R}, i.e. the set of all…

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Q: Let R be a commutative ring. If an # OR and f(x) = ao+a₁x++ana" is a zero divisor in R[x], then an…

A: Given:R is a commutative ring. Also, an≠0R and fx=a0+a1x+...+anxn is a zero divisor in Rx. To…

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 R be a ring. (a) Show that R has exactly one zero, 0R. (b) If R also has an identity, 1R, show…

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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: Let R be a ring with 1. Let 0 a € R. If there exist two distinct elements b and c in R such that ab…

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Q: Abstract Algebra:Prove that the characteristic of a finite ring R divides |R|.

A: The characteristic of a finite ring R divides R

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Q: (a) Give an example showing the following characterisation of a subring is incorrect, and explain…

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Q: Ring theory (abstract algebra)

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Q: Let R be a ring with unity and assume a ∈ R is a unit. Prove that a is not nilpotent.

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

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Q: Let R be a ring. Let S be the set {ab − ba|a, b ∈ R}. Prove that if R is an integral domain, then S…

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Q: Let R be a commutative ring with identity, and let a, b E R. Assume ab is a unit in R. Do a and b…

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