Let R be a euclidean ring and a,b are non zero elements of R.if a and b are associates,show that d(a)=d(b)

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

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Q: Let R be a commutative ring. Then R is an integral domain if and only if ab= ac implies that b = c,…

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Q: Let R be a ring with unity and let a∈R. Prove that if a is a zero divisor, then it is not a unit.

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Q: (a) Let R be a ring and S a subset of R. What does it mean to say that S is a subring of R?

A: a. S is a subset of R. A non-empty subset S of R is a subring if a, b ∈ S ⇒ a - b, ab ∈ S. A subring…

Q: Let R be a commutative ring and let x, y ∈ R. If xy is a zero divisor, show that x or y is a zero…

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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: Let R be a commutative ring. Prove that HO.R (R, M) and M are isomorphic R-modules.

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Q: If Ø: R S is a ring homomorphism. The Ø preserves:

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Q: If a ring R has characteristic zero, then R must have an infinite number of elements. true or false

A: We have find that the given statement  "If a ring R has characteristic zero, then R must have an…

Q: Let R be a Boolean ring with unity e. Prove that every element of R except 0 and e is a zero…

A: Let R be a Boolean ring with unity e. To prove R is a Boolean r ring with unity e, so every element…

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

A: Here you have posted multiple question, So as per the policy I can answer only first question for…

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

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Q: If R is a commutative ring, show that the characteristic of R[x] is thesame as the characteristic of…

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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: 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: 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: 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: Is it true that if S is a unital subring of a unital ring, then the identity elements of the two…

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Q: 4 If R is a commutativ ring, show that the characteristic of R[X] is the same the characteristic of…

A: Given

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: Let R be a commutative ring and a e R. Show that if a is nilpotent, then ab is nilpotent for each be…

A: Solution. Since a is nilpotent , an = 0 for some n ∈ N.

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

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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: If A is a unitary ring define its centre to be CenA = {x €A; Vy EA) xy = yx}. %3D Show that CenA is…

A: Unitary ring is s ring with unity ie  A ring containing multiplicative identity.

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: Is the ring Z/6Z and Z6 are isomorphic.

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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 R is a commutative ring and |R|= 30. If I is an ideal of R and |I| = 10, prove that I is a…

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Q: (a) Let R be a commutative ring with M being maximal ideal in R then R/M is a field. (b) If R is a…

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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: be a Euclidean ring and asb elements of R- gf a and b Let R are non- ze YO asso ciates , Show that…

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Q: Let R be a ring and assume a∈R is not a zero divisor.Prove that if ba=ca,  then b=c.

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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: et R be a ring and assume a∈R is not a zero divisor. If ab=ac for some b,c∈R, then b=c

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Q: Show if R is a commutative ring then R[x] is also a commutative ring.

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Q: Let R be a commutative ring with identity, and let a, b E R. Assume ab is a unit in R. Do a and b…

A: Here given R is a commutative ring with identity. and let a,b∈R and assume ab is a unit. we have to…

Q: Let A be a ring with identity e and let a, b E A such that ab = e. Then ba is a Idempotent element…

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Q: Let A, B,C be rings. Let & be a ring homomorphism from A into B and ß be a ring homomorphism from B…

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Q: Let R be a ring. Let p= ax + b and q = cx² + dx + e be elements of R[x], for a, b, c, d, e E R. True…

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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: Prove that Q[x]/&lt;x2 - 2&gt; is ring-isomorphic to Q[√2] = {a +b√2 | a, b ∈ Q}.

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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: Suppose R is a commutative ring with 1R# 0R. Show that if f (x) = ao + a1a + a2a ++a,n" is a unit in…

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Q: Let R be a commutative ring. Prove that HomR (R, M) and M are isomorphic R-modules.

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Let R be a ring and M be an R-module. Ther

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Q: Let R be a ring such that a6 - = x for all æ E R. Prove that R is commutative.

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Q: If Ø: R → S is a ring isomorphism. The Ø preserves: Units Idempotent elements

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

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