Q: Let be a commutative ring with unity of characteristic 3. Compute and simplify (a + b)° for all…
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Q: 12. Let (I,+,') be an ideal of the ring (R,+, ·). Prove that (I,+, ·) is a primary ideal if and only…
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Q: Let R be a commutative ring and let a ∈ R . Show that I a = { x ∈ R ∣ a x = 0 } is an ideal of R.
A: Given: Let R be a commutative ring and let a ∈ R . To Show that I a = {x∈R ax = 0} is an…
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: Let R be a ring and let I be an ideal of R. Prove that the factor ring R/I is commutative iff rs-sr…
A: We have to prove that factor ring R/I is commutative iff rs-sr is in R for all r and s in R.
Q: Let R be a commutative ring with unity, and let I be a proper idealwith the property that every…
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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: Suppose that K is a commutative ring with identity. If and I are ideals of R for which R/I≈ R/J as…
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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…
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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: Suppose A is a Noetherian unital commutative ring and I is an ideal of A. Prove that A/I is…
A: Some basic results of module theory required to understand the following proof better . If R is a…
Q: Let R be a ring with a subring S. Prove or disprove: If a ∈ R is nilpotent and a ∈ S, then a is also…
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Q: Let R be a commutative ring. If I and P are ideals of R with P prime such that I ¢ P, prove that the…
A: The ideal quotient of P and I is P:I=x∈R : xI⊂P which is again an ideal of R. Given that P is a…
Q: Prove that a ring with unity R has a unique maximal left ideal M if and only if R\M is the set of…
A: Prove that a ring with unity R has a unique maximal left ideal M if and only if R\M is the set of…
Q: Let R be a ring with unity 1R and let S be a subring of R containing 1R. If r∈R is a unit of R and…
A: Let R be a ring with unity 1R and let S be a subring of R containing 1R. If r∈R is a unit of R and…
Q: Let R be a finite commutative ring with identity. Then every prime ideal of R is maximal True O…
A: To prove that every prime ideal of R is maximal.
Q: The ring Z is isomorphic to the ring 3Z O True False
A: Solution:
Q: Let R be a commutative ring with unity and let N= {a e R| a" = 0 for some n e Z"; n> 1} Show that N…
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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: Let R be a commutative ring with unity, and let c ER be a fixed element. (a) Prove that the set A =…
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Q: Let R be a commutative ring with identity and let I be a proper ideal of R. Prove that R/I is a…
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Q: Let R be a commutative ring with identity. Is x an irreducible element of R[x]? Either prove that it…
A: Given that R is a commutative ring with identity.
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: b) Prove that, if S is a ring with characteristic 0, then S infinite.
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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 commutative ring with unity and let a ∈ R be fixed. Prove that the subset Ia = {x ∈ R :…
A: Given below the detailed solution
Q: Let R and S be isomorphism rings. (a) Prove that R ring with unity if and only if S is a ring with…
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Q: Let R be a commutative ring with an identity 1R and let J be a proper ideal with the property that…
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Q: If R is a commutative ring with unity and A is a proper ideal of R, show that R/A is a commutative…
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Q: Let R be a ring with unity and assume a ∈ R is a unit. Prove that a is not nilpotent.
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Q: Let R be a ring with identity. If ab and a are units in R, prove that b is a unit.
A: Since you have asked multiple questions so as per guidelines we will solve the first question for…
Q: Given that (I, t.) in an ideal of the ring (R, +,), show that a) whenever (R,1,) in commutative with…
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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 R be a commutative ring with unity. If I is a prime ideal of R prove that I[x] is a prime ideal…
A: Let R be a commutative ring with unity. If I is a prime ideal of R we have to prove that I[x] is a…
Q: a) If U and V are ideals of a ring R and let UV be the set of all those elements which can be…
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Q: Let be a commutative ring with unity of characteristic 3. Compute and simplify (a + b)6 for all…
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Q: 1. Let R be a commutative ring with unity and let a e Rbe fixed. Prove that the subset Ia = {x E R:…
A: i have provided the detailed proof in next step
Q: 4: prove that Let R be a commutative ring with identity and I be a maximal ideal of R. Then the…
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Q: Let R be a commutative ring with unit element .if f(x) is a prime ideal of R[x] then show that R is…
A: Given R be a commutative ring with unit element. If f(x) is a prime ideal of R[x] then we have to…
Q: If R be a commutative ring with identity and a ∈ R is idempotent different from 0 or 1. Prove that R…
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Q: Let R be a commutative ring such that a^2 = a for all a ∈ R, then show that a+a = 0.
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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: a. Let R and S be commutative rings with unities and f:R -S be an epimorphism of rings. Prove that S…
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Q: Suppose that R is a commutative ring with unity and that I is an ideal of R. Prove that the set of…
A: Given: R is a commutative ring with unity and that I is an ideal of R. To prove: The set of all x∈R…
Q: 3) Let R be a commutative ring with 1. Let A and B be two distinct maximal ideals of R. Show that AB…
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Q: The ring Z is isomorphic to the ring 3Z True False
A: The ring Z has identity 1 as 1·a=a·1=a∀a∈Z The ring 3Z has no identity i.e. there does not exist…
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…
Q: 5. Let A and B be two ideals of a commutative ring R with unity such that A +B = R. Show, that AB=A…
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- 14. Let be an ideal in a ring with unity . Prove that if then .Let I be an ideal in a ring R with unity. Prove that if I contains an element a that has a multiplicative inverse, then I=R.17. Suppose is a ring with positive characteristic. Prove that if is any ideal of then is a multiple of the characteristic of.
- 15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .24. If is a commutative ring and is a fixed element of prove that the setis an ideal of . (The set is called the annihilator of in the ring .)Let R be a commutative ring with unity whose only ideals are {0} and R Prove that R is a field.(Hint: See Exercise 30.)
- Exercises Let be an ideal of a ring , and let be a subring of . Prove that is an ideal ofLet R be a commutative ring that does not have a unity. For a fixed aR, prove that the set (a)={na+ra|n,rR} is an ideal of R that contains the element a. (This ideal is called the principal ideal of R that is generated by a. )Exercises If and are two ideals of the ring , prove that is an ideal of .
- 27. If is a commutative ring with unity, prove that any maximal ideal of is also a prime ideal.Let R be a commutative ring with characteristic 2. Show that each of the following is true for all x,yR a. (x+y)2=x2+y2 b. (x+y)4=x4+y4Assume that each of R and S is a commutative ring with unity and that :RS is an epimorphism from R to S. Let :R[ x ]S[ x ] be defined by, (a0+a1x++anxn)=(a0)+(a1)x++(an)xn Prove that is an epimorphism.