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

Q: Show that the mapping Te data by pla + b/2)=a- bv2, a,beQ is an automorphism of the body Q(/2). Show…

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Q: Construct a ring homomorphism Z|V2| → Z/(7). Verify it is indeed a ring homomorphism (show all…

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Q: The image of a commutative ring under homomorphism may be noncommutative False True

A: As you asked multiple questions , I solved the first one. So you asked "the image of a commutative…

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

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Q: Let F be a field and Ø: F→Fbe a nonzero ring homomorphism, then Ø Is the identity map. Select one:…

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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: Define reducible and irreducible elements in ring R.

A: Remark: An integral domain is a commutative ring with unity which does not contain any…

Q: Prove that the intersection of any collection of subfields of a field Fis a subfield of F.

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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.

A: Solution:

Q: Determine all ring homomorphism from Zn to Zn

A: (if n is not prime) No. of ring homomorphism from Zn to Zn = No. of Idempotent elements in Zn.

Q: Determine all ring homomorphisms from Z to Z.

A: In algebra, A homomorphism is defined as the similarity between the shape, structure, form, etc…

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: 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: Find all possible ring homomorphisms o : Z10 --->Z15 (b) In each case identify the Kern

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Q: Find all ring automorphisms of Q(∛5).

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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: 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: (B) Prove that: 1. Every Boolean ring is commutative. 2. Every field is integral domain.

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

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

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Q: Prove: A field is an integral domain

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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: (3) Let A be commutative ring with identity, then A has just trivial ideals iff A is ........ O…

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Q: Let R = {0, 1, x}, where 0, 1, and x are distinct, and let (R, +, ⋅) be a ring with additive…

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Q: 4) Let : Z--------->Z20 such that (x)= 16x; Show whether is a ring homomorphism or not

A: Given- map ϕ:ℤ→ℤ20 ; ϕ(x)=16x To show whether ϕis a ring homomorphism or not.

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: Q2: Let X be anon-empty set, prove that (p(X), An) is ring?

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Q: is Zs is a field ?if yes show it if not give a reason

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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: 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: Let R be the set of all functions. It is known that (R,+,×) is a ring where + and × are both…

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Q: Prove that the only homomorphisms from Z to Z (Z being the ring of integers) are the identity and…

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Q: Determine whether (Z,⊕,∗) is a ring with binary operations defined as (x⊕y)=x+y−7,…

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Q: determine all ring homomorphisms from Q to Q

A: Determine all ring homomorphisms from Q to Q.

Q: Give an example of a commutative ring without zero-divisors that is not an integral domain.

A: Integral domain: Let D be a ring. Then D is an integral domain, provided these conditions hold: 1. D…

Q: Let x, y, z be elements of the real numbers. Use the field axioms and ordered axioms to prove -0=0.

A: We make use of the following field axioms:

Q: The ring Z,2, has exactly--- --maximal ideals 1 2.

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Q: f:(R,+) (R', +'/) be a ring Homomorphism, and R is integral domain, then R' is integral domain if f…

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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: Prove that the quotient ring is a finite field.

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Q: Show that R[x]/ is a field.

A: In this question, we have to show that ℝxx2+1 is a field.

Q: Prove that the intersection of any collection of subrings of a ring Ris a subring of R.

A: Let S be  intersection of any collection of subrings of ring R. Then we have to prove  S is subring…

Q: Show that (Z,+,.) Is embedded in (Q,+,.).?? Question from rings and fields subject

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Q: Prove that a field has no zero divisors.

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Q: Prove that every field is an integral domain, but the converse is not always true. [IIint: Sce if…

A: Let (F, +, ·) be any field. Therefore F is commutative ring with unity and posses multiplicative…

Q: Let R be a ring with 1. Show that R[x]/{x) ~ R

A: Given that R be a ring with 1 we have to Show that R[x] / <x> ~  R

Angles in Circles

Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.

Arcs in Circles

A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.