Suppose that R is a ring with no zero-divisors and that R contains anonzero element b such that b2 = b. Show that b is the unity for R.

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 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 a commutative ring and let x, y ∈ R. If xy is a zero divisor, show that x or y is a zero…

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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: Suppose that a belongs to a ring and a4 = a2. Prove that a2n = a2 forall n >= 1.

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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: If R is a commutative ring, show that the characteristic of R[x] is thesame as the characteristic of…

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Q: 6. If a and b are not zero divisors in a ring R, prove that ab is not a zero divisor.

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Q: Show that the centre of a ring R is a sub ring of R. And also show that the centre of a division…

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

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Q: Let a and b be elements of a ring. Prove that (-a)b = -(ab).

A: Solve the following

Q: If A and B are ideals of a ring, show that the product of A and B, AB = {a,b, + a,b, +· · · + a,b,l…

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Q: Show that the nilpotent elements of a commutative ring form a subring

A: We have to prove that the nilpotent elements of a commutative ring form a subring. In order to do…

Q: If R is a ring, then every element of R is either a unit or a zero-divisor. Select one: O True O…

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Q: Let U = {a, b}. Define addition and multiplication in P(U) by C +D = CU D and CD = Cn D. Decide…

A: Ring (definition) Let R be a non empty set together with two binary operations called addition(+)…

Q: Find elements a, b, and c in the ring Z ⨁ Z ⨁ Z such that ab, ac, andbc are zero-divisors but abc is…

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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: 21. If R is a ring, prove that R[x] ~R, is the ideal generated by x.

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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: If A and B are ideals of a commutative ring R, define the sum of A and B as: A+B = {a+b| a € A, b €…

A: Let A and B be two ideals of a ring R. If  A and B are ideals of a commutative ring of R with unity…

Q: 4. Prove that a zero divisor in a ring cannot be a unit.

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Q: 5) Suppose that (R, +,.) be a ring without identity and has a subring with identity, then (a) R…

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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: For any element a in a ring R, define (a) to be the smallest ideal of R that contains a. If R is a…

A: An ideal is a non-empty sub set I of a ring R, such that

Q: In a ring R without unity, show that every idempotent is a zero divisor but is not nilpotent.

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Q: If R is a noncommutative ring with unity and x, y ∈ R, compute the product x(x + y)(x − y)y.

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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: Suppose that a and b belong to a commutative ring and ab is a zero-divisor.Show that either a or b…

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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: Suppose that R is a commutative ring with unity 1. Then if ab is a zero divisor then a or b is a…

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Q: Let R be a commutative ring. Show that R[x] has a subring isomorphicto R.

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Q: If A and B are ideals of a commutative ring R with unity and A + B= R, show that A N B = AB.

A: Given Data: A and B are ideal of a commutative ring R with unity And A + B = R   For A⋂B = AB, The…

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 that if n is an integer and a is an element from a ring, thenn . (-a) = -(n . a).

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Q: Let R be a commutative ring, a, b e R and ab is a zero-divisor. Show that either a or b is a…

A: We have to solve given problem:

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: Find two elements a and b in a ring such that both a and b are zerodivisors,a + b ≠ 0, and a + b is…

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Q: Let A be a ring with identity e and let a, b E A such that ab = e. Then ba is a Idempotent element…

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Q: Show that if R is a ring with unity and N is an ideal of R such that N ≠ R , then R / N is a ring…

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Q: If R is a commutative ring, shw that the characteristic of R[x] is the same as the characteristic of…

A: Given: R is commutative ring

Q: Suppose that R is a commutative ring without zero-divisors. Showthat all the nonzero elements of R…

A: Given: R is a commutative ring without zero-divisors. To show: all the nonzero elements of R have…

Q: elements

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Q: Let a and b be elements in a ring R. If ab is a zero divisor, prove that either a or b is a zero…

A: Given :          ab is a zero divisor where a and b are elements in ring R. Formula :        Zero…

Q: Show that the factor ring Q[r] (P(x)) has a zero-divisor

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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: Suppose that R is a commutative ring with unity and that I is an ideal of R. Prove that the set of…

A: Given: R is a commutative ring with unity and that I is an ideal of R. To prove: The set of all x∈R…

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: Indicate such a subring of the ring P[x], which contains P and is different from P, but is not…

A: Image is attached with detailed solution.

Q: Is the set of all real numbers of the form a + b5, where a and b are rational numbers a ring with…

A: To check A=a+b53:a,b∈Q under usual addition + and usual multiplication × is a ring or not. Ring: A…