Let R be a ring with 1. Show that R[x]/{x) ~ R
Q: Let R be a commutative ring and let a ∈ R . Show that I a = { x ∈ R ∣ a x = 0 } is an ideal of R.
A: Given: Let R be a commutative ring and let a ∈ R . To Show that I a = {x∈R ax = 0} is an…
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 (?,+, ⋅) be a ring with additive identity 0. Prove that for all x∈?, 0⋅x=0 and x⋅ 0 = 0.
A: We know that if (R,+,.) is a ring then (R,+) is an abelian group. And in abelian group, cancellation…
Q: Let R be a ring with a finite number n of elements. Show that the characteristic of R divides n.
A: Given : R is a ring with n elements. To prove : The characteristic of a ring R divides the number…
Q: Let S be a ring. Determine whether S is commutative if it has the following property: whenever æy =…
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Q: Let R be a ring with unity 1. Show that S = {n· 1 | nE Z} is a sub- ring of R.
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Q: Let u be a unit in a ring R. Show that u divides x, for all x in R. (that is, show that x uy for…
A: Given that u be a unit in a ring R. Then by the definition the non zero element u has a…
Q: (B) Let I be a maximal proper ideal of commutative ring with identity R. Prove that R/I is a field.
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Q: Let R be a ring with identity 1 and let a be an element of R such that a2 = 1. Let S = { ara : r e…
A: we will use sub ring test.
Q: Let R be a commutative ring with 10. Prove that R is a field if and only if 0 is a maximal ideal.
A: If R is a field, then prove that {0} is a maximal ideal. Suppose that R is a field and let I be a…
Q: Let R be a Boolean ring with unity e. Prove that every element of R except 0 and e is a zero…
A: Let R be a Boolean ring with unity e. To prove R is a Boolean r ring with unity e, so every element…
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: Let R be a commutative ring. Prove that HomR(R, M) and M are isomorphic R-modules
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Q: Indicate such a subring of the ring P[x] that contains P and is different from P but is not…
A: There are so many examples can be found.
Q: 1. Let R be a ring with the additive identity 0. Prove that for any a E R, 0- a = 0.
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Q: consider the mapping from M2(z) into Z,prove or disprove that this is a ring homomorphism
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Q: 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…
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…
Q: Let R be a ring and S be a subring of R with OS, OR being the zero elements in S, R respectively.…
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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...
Q: If is a homomorphism from the ring R to the ring R' , show that; a) (0)=0 b) (−r)= −(r)for all…
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Q: Let R be a ring with a multiplicative identity 1R. Let u, an element of R, be a unit. Prove: u is…
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Q: For a fixed element a of a ring R, prove that the set {x ϵ R I ax = O} is a subring of R.
A: Given : The ring R and a fixed element 'a' of R. To prove that the set x ∈ R | ax =…
Q: Let R be a commutative ring with identity. Is it possible for R[x] to be a PID without being a…
A: Yes , it is possible for R[x] to be a PID ( assuming R[x] is PID ) without being a Euclidean domain.
Q: Q1: Let R be a commutative ring with Char(R) = 2 and let p:R → R be defined such that o (a) = a².…
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Q: Let S be a ring. Determine whether S is commutative if it has the following property: whenever ry =…
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Q: Let R be a ring with m elements. Show that the characteristic of Rdivides m.
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Q: What does the notation R* mean with R being a ring with unity? Let R be a ring with a subring S:…
A: What does the notation R* mean with R being a ring with unity? Let R be a ring with a subring S:…
Q: Let R be a commutative ring with identity. Is x an irreducible element of R[x]? Either prove that it…
A: Given that R is a commutative ring with identity.
Q: Let R be a ring with a subring S: Prove or disprove: If a ∈ R is a unit, and a ∈ S, then a is also a…
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Q: Let R be a finite ring and α ∈ R with α ≠ 0. If α is not a zero divisor, then α is a unit.
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Q: Let R be a ring and a=a for all a'e R, Then commutative. prove that R is
A: First we notice that x3=x for all x∈ℝ, so that means 2x3=2x and thus 8x=8x3=2x and so 6x=0. Thus…
Q: Let R be a commutative ring. Show that R[x] has a subring isomorphicto R.
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Q: 5. Let F be a field and 0 : F → R be a ring epimorphism. If Ker0 + F, show that R has no zero…
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Q: let (Z,+,*) be a ring of integer number and (Ze,+,*) is ring of even integer number and f:Z→Ze such…
A: Given : (Z,+,*) is a ring of integer numbers. (Ze,+,*) is a ring of even integer numbers. To…
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: Let (R, +, .) be a nontrivial ring with * identity, prove that 1 0
A: It is given that (R,+, .) be a nontrivial ring with identity. Now we have to show that 1≠0. So, (R,…
Q: Let R be a commutative ring of characteristic 2. Prove that : (a+ b) = a² +b² = (a - b)? v a, be R.…
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Q: Let R be a ring with identity. If ab and a are units in R, prove that b is a unit.
A: Since you have asked multiple questions so as per guidelines we will solve the first question for…
Q: Let (R,+, ⋅) be a ring with additive identity 0. Prove that for all x∈R, 0⋅x=0 and ? ⋅ 0 = 0.
A: Solution
Q: If I is an ideal of a ring R, prove that I[x] is an ideal of R[x].
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Q: Let f:R→S be a ring homomorphism. (i) Prove that if K is a subring of R then fIK) is a subring of s-…
A: Suppose f:R→S be a ring homomorphism then ; fx1+x2=fx1+fx2 for all x1,x2∈R. fx1·x2=fx1·fx2 for all…
Q: Suppose I, J be ideals of a commutative ring R. Prove that IJ cIn).
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Q: The map f: Z→ Z,o given by f(x)= 2x is a ring homomorphism. Select one. True False
A: SINCE YOU HAVE ASKED MULTIPLE QUESTIONS IN SINGLE REQUEST, WE WILL BE ANSWERING ONLY THE FIRST…
Q: Let I be a maximal proper ideal of commutative ring with identity R. Prove that R/I is a field.
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Q: Let R be a commutative ring such that a^2 = a for all a ∈ R, then show that a+a = 0.
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Q: Let R be a commutative ring. Prove that HomR (R, M) and M are isomorphic R-modules.
A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…
Q: Let R be a ring such that a6 - = x for all æ E R. Prove that R is commutative.
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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.
A: We are given that R be a commutative ring with unity. We have to show that R is a field if and only…
Q: Suppose that R and S are isomorphic rings. Prove that R[r] = S[r].
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Q: If µ is finitely additive on a ring R; E, F eR show µ(E) +µ(F) = µ(Eu F)+µ(En F) %3D
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- Let R be a commutative ring with characteristic 2. Show that each of the following is true for all x,yR a. (x+y)2=x2+y2 b. (x+y)4=x4+y4Let R be a commutative ring with unity whose only ideals are {0} and R Prove that R is a field.(Hint: See Exercise 30.)Let I be an ideal in a ring R with unity. Prove that if I contains an element a that has a multiplicative inverse, then I=R.
- 22. Let be a ring with finite number of elements. Show that the characteristic of divides .12. Let be a commutative ring with prime characteristic . Prove, for any in that for every positive integer .17. Suppose is a ring with positive characteristic. Prove that if is any ideal of then is a multiple of the characteristic of.
- Exercises Let be an ideal of a ring , and let be a subring of . Prove that is an ideal of[Type here] 23. Let be a Boolean ring with unity. Prove that every element ofexceptandis a zero divisor. [Type here]24. If is a commutative ring and is a fixed element of prove that the setis an ideal of . (The set is called the annihilator of in the ring .)