Let a, b, and c be elements of a commutative ring, and suppose thata is a unit. Prove that b divides c if and only if ab divides c.

Q: Let R be a euclidean ring and a,b are non zero elements of R.if a and b are associates,show that…

A: Let R be a Euclidean domain with measure d. Also given that a and b are associates in R. Then from…

Q: if in a ring R every x in R satisfies x^2=x , prove that R must be commutative

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Q: а. Let a, b, c be three elements of a Euclidean ring R and (a, b) = 1. If a/bc then show that a/c.

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Q: 6. Suppose that (M,+,.) be a maximal ideal of the commutative ring with identity (R, +,.) and x E M,…

A: Since, given that (M, + .) Is a maximal ideal of the commutative ring with identity (R, +, .)  and…

Q: * (10 Let (R, +,) be commutative ring with identity. Then a - b = a-c-b=c ifand only if R has no…

A: True

Q: The number of idempotents elements in the ring Zs is: O 2 4 8 1 O O O

A: Ans : 4  Option 2nd true

Q: Iffis a ring homomorphism from Zm to Zn such that f (1) = b, then bak+2 = bk. True O False

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Q: Prove that if a is a ring idempotent, then an = a for all positive integers n.

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Q: Let a and b be idempotents in a commutative ring. Show that eachof the following is also an…

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Q: A ring (R. +.) .) is commutative if addition is commutative in R. O True O False

A: Solve the following

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: Decide whether ZxZ = {(n,m) n,me Z} with addition and multiplication by components, give a ring…

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Q: Iffis a ring homomorphism from Zm to Zn such thatf(1) = b, then bak+2 = bk. True False

A: Ring homorphism

Q: If R is a commutative ring and Ø:R→S is a ring homomorphism, then S is a commutative ring True O…

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Q: If A, B, C are ideals of a ring R such that BCA, prove that An (B + C) = B+(AnC) = (A B) + (AnC).

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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 a is an idempotent element of a commutative ring, then which one of the following is also an…

A: From given statement

Q: Any ring R is a Jacobsen radical ring, if it is a simple ring. O مبا O

A: JACOBSON RADICAL:- The radical of the base ring R is called its Jacobson radical and denoted by…

Q: Let R = {2n: n E Z} and define addition and multiplication O in R by a b = a + b and aOb = for all…

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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 CR,t,,) be a Commutative ring ?

A: First I recall you Definition of commutative ring.  A commutative ring is ring in which…

Q: Let R and S be commutative rings. Prove that (a, b) is a zero-divisorin R ⨁ S if and only if a or b…

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Q: The number of idempotents elements in the ring Zs is: O 1 O 2 O 8 O 4

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Q: Label each of the following statements as either true or false.   If one element in a ring R has a…

A: Label each of the following statements as either true or false.   If one element in a ring R has a…

Q: prove that the rings (R,+,.) and (Q,+,.) are fields.

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Q: Let R be Euclidean ring and a, b non - Zevo elements of R- If dca) <d lab), are show that b is not…

A: Let a and b be non-zero elements of R. Assume that b is a unit in R. Then, for some element c in R…

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: 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: A ring, R, in which r^2 = r for all elements r in R is called a Boolean ring. In any Boolean ring,…

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Q: The number of zero divisors of the ring Z4 O Z, is

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Q: Prove that if u is a unit in a ring R, then -u is a unit in R.

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Q: Iffis a ring homomorphism from Zm to Zn such thatƒ(1) = b, then b4k+2 = bk. %3D True False

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Q: Iff is a ring homomorphism from Zm to Zn such thatf(1) = b, then bak+2 = bk . O True O False

A: False

Q: . If A, B and Ç are ideals of a ring R, prove that A (B+ C) = AB+AC.

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Q: if A, B, C are ideals of a ring R such that BCA, prove that An (B + C) = B+ (AnC) = (A B) + (AnC).

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Q: 9. Suppose that (R,+,.) be a commutative ring with identity and x E rad R, then (a) (x) = R (b) 1 —…

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Q: If R is a commutative ring and Ø:R→S is a ring homomorphism, then S is a commutative ring * O True O…

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Q: If A, B, C are ideals of a ring R such that BCA, prove that An (B + C) = B+ (AnC) = (An B) + (AnC).

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Q: Let A be a commutative ring with identity and D be an integral domain. Suppose that p: A → D is a…

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Q: Iffis a ring homomorphism from Zm to Zn such that f(1) = b, then b4k+2 = bk. True False

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Q: Prove that if u is a unit in a ring R, then u is a unit in R. -

A: According to the given information, It is required to prove that if u is a unit in a ring R then -u…

Q: Suppose I, J be ideals of a commutative ring R. Prove that IJ cIn).

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Q: Let R be a commutative ring with more than one element. Provethat if for every nonzero element a of…

A: We have given commutative ring R with more than one element. We have to prove R has a unity and…

Q: Determine all elements of a ring that are both units and idempotents

A: To determine: All elements of a ring that are both units and idempotent.

Q: Label each of the following statements as either true or false. Any ring with unity must be…

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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 ring and M be an R-module. Ther

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Q: Verify that the following structures form a ring. (a) The set Z[ √ 2] of real numbers of the form…

A: The given set is Z2 of real numbers of the form m+n2 where m,n are integers with the usual addition…

Q: The number of idempotents elements in the ring Zo is: 1 2 8 O 4

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Q: Va, beZatb = a + b +2 and aob = a tabtb is a Ring!

A: