5. An element x in a ring R is called idempotent if a2 = x. Prove that if a is an idempotent element in a ring R with unity e, then e - a is also idempotent.

Q: If Z is integers ring, then each non zero element in a proper ideal of Z is......... O (i)…

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Q: disprove that the is smallest non- Prove commutative ring oY of order 4-

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

A: We know that if (R,+,.) is a ring then (R,+) is an abelian group. And in abelian group, cancellation…

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: Suppose that Φ: R --> S is a ring homomorphism and that the imageof Φ is not {0}. If R has a…

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Q: 7. Suppose that (R, +..) be a commutative ring with identity and (I, +,.) be an ideal of R. If I is…

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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 a ring is commutative if it has the property that ab = caimplies b = c when a ≠ 0.

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Q: Show that a homomorphism from a field onto a ring with more thanone element must be an isomorphism.

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Q: If a is an idempotent in a commutative ring, show that 1 - a is alsoan idempotent.

A: Here given

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

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Q: The number of idempotents elements in the ring Z6 is: 8 4 1

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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: Show that Ø is a ring homomorphism.

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Q: 1- Let ý:R, » R, be a ring homomorphism such that Kerø =. Then, o is a) 1-1 b) onto c) Both 1-1 and…

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Q: 13. Let Rbeacommutative ring. If a Og and f(x) = t ajx+ ayx +h (with a,O) is a zero divisor in A,…

A: In a ring R,if ab=0 implies a,b are not zero then a and b are zero divisors in R

Q: 2- Let f be an isomorphism from the ring (R, +,) to the ring (R', +','). If (I, +;) is an ideal of…

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Q: 1- Let ý:R, » R, be a ring homomorphism such that Kerø = . Then, ø is a) 1-1 b) onto c) Both 1-I and…

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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: 4 IF Risa Commutativ ring, Show that the characteristic of REX] is the Same as the characteristicof…

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Q: Construct the adition and multiplication tables for the ring Z/4Z.

A: Given: A ring ℤ/4ℤ. To construct: Addition and multiplication tables for the ring ℤ/4ℤ.

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: Show that the ring (2ot) is principal idenl ring.

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Q: Show that the subring of an artinain ring may not be artinian???

A: According to the given information, it is required to show that subring of an Artinian ring may not…

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

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Q: Explain why the ring of integers Z under usual addition and multiplication is not a field.

A: Given that the set of integers Z is a ring under addition and multiplication. We know that an…

Q: Q1: Let S, and Szare two subrings of a ring (R, +,.), prove that S, USz is subring of R iff either…

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

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Q: Prove or dlspive that the subset M: ta+3biabeZ5 IS a subring of the Causslan integer ring ZCiJ

A: If R is a ring  and S is a subset of R is said to be a  subring if it is closed in the following…

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: 7. Suppose that (R, +,.) be a commutative ring with identity and (I, +,.) be an ideal of R. If I is…

A: Prime Ideal:- An ideal P of a commutative ring R is prime if it has the following two properties:…

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

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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: (1) If Z is integers ring, th each non zero element in proper ideal of Z is........ (i) invertible O…

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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: 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: 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: 4) In a ring R; The sum of two non-trivial idempotent elements is not always an if we take + = is…

A: 4) We need to show that , sum of two non trivial idempotent elements is not always idempotent. We…

Q: 1. An element a of a ring Ris called an idempotent if a? = a. Show that a field contains exactly two…

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Q: The number of idempotents elements in the ring Z6 is: 1 8 4

A: Ans : 4  Last option is true List of idempotent element is {0,1,3,4}

Q: 17. Given that (I, 1.) in an ideal of the ring (R,+,), show that a) whenever (R,1,) is commutative…

A: Definition Two-sided Ideal of a ring : An ideal I of a ring R is called a two-sided ideal if it…

Q: Show that the rings (3Z/60Z)/(12Z/60Z) and 3Z/12Z are isomorphic. Then show tha both isomorphic to…

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