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

Q: Q1) In a ring Z, find the least common multiple of {2,3,8,12}.

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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: If the ring R has a left identity as Well right identity then these two as are equal.

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Q: The ring 5Z is isomorphic to the ring 6Z OTrue O False

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Q: In the ring Z;[x]': 1- The inverse of x is:

A: Answer :

Q: Show that 7 is irreducible in the ring Z[V5].

A: Consider the ring ℤ5. Any element of the above ring is of the form a+b5 where a and b are integers.…

Q: The number of zero divisors of the ring Z, Z5 is О 1 O 5

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: 3. Prove that an ideal I in a ring R is the whole ring if and only if 1 e I.

A: Question: Prove that an ideal I in a ring R is the whole ring if and only if 1∈I. Proof: We have to…

Q: If R is a commutative ring with unity and a e R, then (a) = {ar : reR}=aR.

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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: - Prove that, if I is an ideai of the ring Z of integer numbers then I=, for some nɛZ'U{0}

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

A: Solve the following

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

A: Here, given that The ring R with the cancellation law for multiplication holds in R. Let a,b,c∈R if…

Q: The number of nilpotent elements in the ring Z23 is: 1 4

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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: The measure u is monotone on the ring. So that µ(A) < µ(B) if ACB

A: Given that The measure is monotone ob the ring,  So we need to consider the following;

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: Q1) In a ring Z, find the least common multiple of {2,3,8,12).

A: Given problem Given that   In a ring Z , We have to find the Least common multiple of { 2,3,8,12…

Q: Rp { -b a | a, b = Zp}

A: It is given that,  Rp = ab-ba : a, b ∈ ℤp We have to prove that,  Rp is a commutative ring with unit…

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: The number of nilpotent elements in the ring Z23 is

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: The set of all units of the ring Zg is

A: SOLUTION: The set of all units of the ring Z8={0,2,4,6} because, f(0)=f(2)=f(4)=f(6)=0

Q: Let R be a ring with m elements. Show that the characteristic of Rdivides m.

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Q: Determine all reversible elements in ring Z_6 and Z_8 and give their inverse

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Q: The set of all zero divisors of the ring Z6 is اختر احدى الاجابات O {2, 3, 4} O (1, 3, 5) O {1, 2,…

A: Z6={0,1,2,3,4,5}Since 2 · 3 ≡6≡ 0 (mod 6) and 3 · 4 ≡12≡ 0(mod 6)However, 1 and 5 are not zero…

Q: Find the charactes ist c of the ring Z2 Zg

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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: The number of zero divisors of the ring Z4 Z2 is O 5 O 1

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

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Q: Prove that the number i5 is not reversible in the ring Z[V-5]

A: Here we show that isqrt(5) is not reversible in the ring Z[sqrt(-5)].

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 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: 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: 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 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: Consider the ring (Q,Ð,0), where the "addition" O and "multiplication" ® are defined as follows:…

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Q: 8. List all ideals of the ring Z12-

A: Ideals of the ring Z12

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: The number of nilpotent elements in the ring Z23 is:

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Q: For any n E N, each zero divisor in the ring Z, is nilpotent.

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Q: In the ring Z[x]/: 1- The inverse of x is:

A: The given ring is ℤ2x/x3+1. The objective is to find the inverse of x and to find Uℤ⊕ℤ.

Q: The number of idempotents elements in the ring Z6 is: O 2 O 8

A: The idempotent element in the ring Z6 is 4 .   An element e in a ring R is said to be an idempotent…

Q: Show, by example, that for fixed nonzero elements a and b in a ring, the equation ax = b can have…

A: Solution:-           consider the ring Z4         Let a=2 &amp;  b=2:         Then 21=2 &amp;…

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

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Q: The ring Z,3 has exactly------------maximal ideals 3 4 1 O O O O

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Q: An element x in a ring is called an idempotent if x2 = x. Prove that  the characterstic of R is 0 or…

A: An element x in a ring is called an idempotent if x^2 = x

Q: Consider the ring R = [r, s,t) whose addition and multiplications tables are given below, Then ts S…

A: Given R=r,s,t be the ring under addition and multiplication. Given table is +rstrrstsstrttrs We have…

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…

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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