Let R be a commutative ring with unity and let N={ aER | a"=0 for nez*, n>1}. Show that N is an ideal of R.

Q: 37. An element x in a ring is called an idempotent if x2 = x. Prove that the only idempotents in an…

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Q: Let be a commutative ring with unity of characteristic 3. Compute and simplify (a + b)° for all…

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Q: Let R be a commutative ring. Then R is an integral domain if and only if ab= ac implies that b = c,…

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Q: Let R be a commutative ring with unity, I an arbitrary ideal in R and P1,P2,...,Pn prime ideals in…

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Q: Let R be a commutative ring with 1 and I be a proper ideal of R. Prove that I is prime if and only…

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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 R be a ring with unity which has exactly one maximal left ideal M. Show that the only…

A: Given: R is a ring with unity which has exactly one maximal left ideal M. To show: The only…

Q: Suppose that R and S are commutative rings with unities. Let ø be a ring homomorphism from R onto S…

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Q: If a ring R has characteristic zero, then R must have an infinite number of elements. true or false

A: We have find that the given statement  "If a ring R has characteristic zero, then R must have an…

Q: Verify in the following statement involving the ideal generated by (a1, a2 , ... , ak ) in the ring…

A: To show that 2,3=ℤ Let 2,3=2x+3y: x,y∈ℤ -------(1) gcd2,3=1 , Thus 2m+3n=1 for some…

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: Verify in the following statement involving the ideal generated by (a1, a2, ... , ak )) in the ring…

A: To prove that: (2, 5)=ℤ

Q: (3) Let A be commutative ring with identity, then A has just trivial ideals iff A is ....... O…

A: Here you have posted multiple question, So as per the policy I can answer only first question for…

Q: Suppose that (R, +, . ) is a commutative ring with identity and it has nontrivial ideals, then R…

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Q: Suppose that R is a commutative ring and |R| = 30. If I is an idealof R and |I| = 10, prove that I…

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Q: 21. A ring is said to be a local ring if it has a unique maximal ideal. If (R,+,) is a local ring…

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Q: Find all possible ring homomorphisms o : Z10 --->Z15 (b) In each case identify the Kern

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Q: Is it true that if S is a unital subring of a unital ring, then the identity elements of the two…

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Q: 4 If R is a commutativ ring, show that the characteristic of R[X] is the same the characteristic of…

A: Given

Q: 18. Let (R, ,) be a commutative ring with identity and let N denote the set of nilpotent clements of…

A: a). As given that N is nonempty as 0∈N.  an=0 (n= positive integer) Then r∈R ran=rnan=0 Let take a,…

Q: Prove that in a ring R having exactly one maximal ideal M, the only idempotents are 0 and 1.

A: Assume R is local and let M=R∖Rx. By assumption M is an ideal. It is also maximal because any ideal…

Q: Let QEX] be the ring of al4 polynomfals with rational coefficients. use the First Isomorphism…

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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: 24. Let (R, +,) be a commutative ring with identity and a ER be an idempotent which is different…

A: R, +,· is said to be commutative ring if Suppose R is a non empty set such that for any two elements…

Q: Let R be a ring with unity 1. If 1 has infinite order under addition then the characteristic of R is…

A: The order of an element a in a ring R is the least positive integer n such that n·a=0. If no such…

Q: Prove that a nonzero commutative ring with unity R is a field if and only if it has two ideals (0)…

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Q: (3) Let A be commutative ring with identity, then A has just trivial ideals iff A is ........ O…

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Q: Let R be a finite commutative ring with unity.  Show that every prime ideal in R is a maximal ideal.

A: We have to show that if R be a finite commutative ring with unity then every prime ideal in R is a…

Q: Let R = {0, 1, x}, where 0, 1, and x are distinct, and let (R, +, ⋅) be a ring with additive…

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Q: (a) Let R be a commutative ring with M being maximal ideal in R then R/M is a field.

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Q: Let (R,+,.) be acommutive ring with identity and a∈R be an idempotent which is different from 0 or 1…

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Q: 7. a) Prove that every field is a principal ideal ring. b) Consider the set of numbers R {a+ bV2|a,…

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Q: IN denotes the set of noninvertible ele conditions are equivalent: (N,+,) is an ideal of (R,+, ), p)…

A: Given N denote the set of non-invertible  elements of R.

Q: Prove that the numbers 3 and 7 are indecomposable in the ring Z[V-5], but the number 5 is…

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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: 2. Let R be a commutative ring with unity. If I is a prime ideal of R, prove that I [x] is a prime…

A: Given R be a commutative ring with unity and if I is a prime ideal of R. Then we have to prove that…

Q: Let (R,+, ⋅) be a ring with additive identity 0. Prove that for all x∈R, 0⋅x=0 and ? ⋅ 0 = 0.

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Q: If Ris a ring with identity and a is a unit, prove that the equation ax = has a unique solution in…

A: Let R be a ring with identity and a∈R be a unit. Prove that the equation ax=b has a unique solution…

Q: Let be a commutative ring with unity of characteristic 3. Compute and simplify (a + b)6 for all…

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Q: The cancellation laws for multiplication are satisfied in a ring R, if R has zero divisor.

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Q: Find all ring homomorphisms from R to Z5.

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Q: 4: prove that Let R be a commutative ring with identity and I be a maximal ideal of R. Then the…

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Q: A ring element a is called an idempotent if a2 = a. Prove that theonly idempotents in an integral…

A: Definition: A commutative ring with unity is called integral domain if a·b=0  ⇒  a=0   or  b=0.

Q: If R be a commutative ring with identity and a ∈ R is idempotent different from 0 or 1. Prove that R…

A: Given that R is a commutative ring  with unity. Also, a∈R  is an idempotent element ⇒a2=a ( a≠0,1)…

Q: Let R be a commutative ring with unity  If I is a prime ideal of R, prove that I[x] is a prime…

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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: 3) Let R be a commutative ring with 1. Let A and B be two distinct maximal ideals of R. Show that AB…

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Q: is union of two ideal rings R, an idea of R? prove or give counter example

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Q: 21. A ring is said to be a local ring if it has a unique maximal ideal. If (R,+, ) is a local ring…

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Q: 5. Let A and B be two ideals of a commutative ring R with unity such that A +B = R. Show, that AB=A…

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