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

Q: Show that a unit of a ring divides every element of the ring.

A: Let us assume a unit in a ring R as a. Let us assume any element as x in that ring R.  By using the…

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

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Q: Show that in the factor ring Z[x] /(2x+1), the element x+(2x+1) is a unit.

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Q: The number of zero divisors of the ring Z4+Z4 is?

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Q: Prove that a division ring has no zero divisors.

A: Answer: Proof:

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: Let R be a ring with unity. Show that {а)3D( 2 хау: х, у є R}. finite

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

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Q: The number of zero divisors of the ring Z4 O Z3 is

A: We have to find the number of zero divisors in the ring Z4⊕Z3.

Q: a is a unit in a ring R with unity, then a is not a zero divisor

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

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Q: Let R is ring of real numbers and *, (2) O defind on R^2 as follows, (a, b), (c, d)ER^2 then (a, b)*…

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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 with…

A: Since we have given that R is a ring with unity and N is a proper  ideal of R . We prove R/N is a…

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: 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: 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: Show that the set 4Z U 5Z is not a subring of the integer ring Z:

A: Z = { ....., -3, -2, -1, 0, 1, 2, 3, ..... }. 4Z = { ...., -8, -4, 0, 4, 8, .... }. 5Z = { ....,…

Q: Let R is ring of real numbers and *, (2) O defind on R^2 as follows, V (a, b), (c, d)ER^2 then (a,…

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Q: 5. An element x in a ring R is called idempotent if a2 = x. Prove that if a is an idempotent element…

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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: prove or disprove that the smallest non commutative ring is of order 4

A: Using the Result : " All rings of order p2 ( p is any prime) are commutative"   As 4=22 and 2 is a…

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: 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: 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: 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: Prove or disprove that Q(√3) and Q(√-3) are ring-isomorphic.

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Q: lisprove: that the triple <Z,,, constitutes a ring. et of units U(Z₂), and state whether (U(Z₁), Ⓒ)…

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

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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: Show that in a ring R cancellation laws hold if and only if R has no zero divisors.

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Q: Suppose that R is a ring with no zero-divisors and that R contains anonzero element b such that b2 =…

A: Given Suppose that R is a ring with nonzero-divisors and that R contains a nonzero element b such…

Q: Show that a commutative ring R with unity is a field if and only if its only two ideals are (0) and…

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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- x…

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

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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: 8. List all ideals of the ring Z12-

A: Ideals of the ring Z12

Q: Prove that a finite ring R with unity and no zero divisors is a division ring.

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

A: The set of all zero divisors of the ring Z6 is (0,2,4)

Q: 14) Any subring of a commutative ring is commutative. True False

A: I have given an answer in step 2.

Q: Prove that the set of all elements in a ring R that are not zero divisors is closed under…

A: We will prove that the set of all elements in a ring R that are not zero divisors is closed under…

Q: 5. Give an example where a and b are not zero divisors in a ring R, but the sum a +b is a zero…

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

A: