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

Q: Let R be a ring with unity 1 and char (R) = 4. %3D Then R contains a subring isomorphic to Q ZO Z3 O

A: IN the given question,  Given that: R is a ring with unity 1 and char(R)=4. we have to find: we have…

Q: Let R be a euclidean ring and a,b are non zero elements of R.if a and b are associates,show that…

A: Let R be a Euclidean domain with measure d. Also given that a and b are associates in R. Then from…

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 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: Let K be a commutative ring. For every AÃ ÃA = det (A) In- = quence, A is invertible iff det(A) is…

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

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Q: Let R be a ring with unity. Show that (a) = { £ xay : x, y e R }. finite

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Q: Let R be a ring with unity 1. Show that S = {n· 1 | nE Z} is a sub- ring of R.

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Q: Let R be a ring with unity in which r^2 = r for every r in R. Prover that if a does not equal 0 and…

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Q: Find an example of a commutative ring A that contains a subset, say S, such that for every s E S we…

A: We can find n commutative ring A such that A contains a subset S such that for every s in S we have…

Q: Iffis a ring homomorphism from Zm to Zn such that f (1) = b, then bak+2 = bk. True O False

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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: Iffis a ring homomorphism from Zm to Zn such thatf(1) = b, then bak+2 = bk. True False

A: Ring homorphism

Q: Let R be a ring such that a² = a for all a E R, and assume that R has an identity. Show that the…

A: Given R be a ring with unity 1 such that a2=a, for all a∈R. Let b∈R be a unit in R. Therefore b-1…

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

A: Solve the following

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: The ring Z is isomorphic to the ring 3Z O True False

A: Solution:

Q: If is a homomorphism from the ring R to the ring R' , show that; a) (0)=0 b) (−r)= −(r)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: Let R be a ring with a multiplicative identity 1R. Let u, an element of R, be a unit. Prove: u is…

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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: Let R be a ring with unity, n a positive integer and a, b e R. Prove: If ab = ba, then (a + b)"…

A: Mathematical Induction  Let us consider a statement P(n) 1. Prove the given statement form n=1…

Q: Let R and S be rings and let ø : R → S be a ring homomorphism. Show that o is one-to-one if and only…

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

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Q: Let a be an element of a ring R such that a3=1R. Prove: for any integer n, either (an)n=1R or…

A: Let a be an element of a ring R such that a3=1R.  We will find, for any integer n, either (an)n is,…

Q: Let R be a ring with unity and assume p, q, r ∈ R∗ Find (pqr)-1and also prove that it’s the inverse…

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Q: Let R be a commutative ring with identity. Is it possible for R[x] to be a PID without being a…

A: Yes , it is possible for R[x] to be a PID ( assuming R[x] is PID ) without being a Euclidean domain.

Q: Q1: Let R be a commutative ring with Char(R) = 2 and let p:R → R be defined such that o (a) = a².…

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Q: Let R be a ring and a=a for all a'e R, Then commutative. prove that R is

A: First we notice that x3=x for all x∈ℝ, so that means 2x3=2x and thus 8x=8x3=2x and so 6x=0. Thus…

Q: Let R be a commutative ring. Show that R[x] has a subring isomorphicto R.

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Q: 1. Let M be a proper ideal in a Boolean ring R with unity. Prove that (i) R/M is a Boolean ring and…

A: It is given that, M is a proper ideal in a Boolean ring R with unity.

Q: Let R be a ring with unity 1, and S = {n.1 : n E Z} . Then S'is Ra subring of Rnot a subring of

A: Let, x ,y in S. So, x = n•1 and, y = m•1 for some n, m in Z.

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

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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: 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: Let R be a ring with unity 1, and S = {n.1 : n E Z}.Then S'is Ra subring of Rnot a subring of

A: A non-empty set R with two binary operations addition(+) and multiplication(·) is called ring if it…

Q: Let M be a proper ideal in a Boolean ring R with unity. Prove that (1) R/M is a Boolean ring and…

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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: Iffis a ring homomorphism from Zm to Zn such that f(1) = b, then b4k+2 = bk. True False

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Q: Show that the factor ring Q[r] (P(x)) has a zero-divisor

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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: Let I be a maximal proper ideal of commutative ring with identity R. Prove that R/I is a field.

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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: (b) Show that if : R→ S is only assumed to be a ring homomorphism, then it is possible to have a…

A: We can prove the above result using two contradictory examples. Note that these examples are only…

Q: 3. Let R be any commutative ring with unity, and let T[r] be the subset of all polynomials with zero…

A: Given that R is a commutative ring with unity, and T[x] be a the subset of all polynomials with zero…

Q: Let R be a ring with unity e. Verify that the mapping θ: Z---------- R defined by θ (x) = x • e is a…

A: Let R be a ring with unity e, verify the mapping θ:Z→R defined by θx=x.e is a homomorphism If R and…