Let R be a commutative unital ring. Let I be an ideal of R. Denote by (I) the ideal in R[a] generated by the image of elements in I under the natrual embedding incl : R → R[x]. Prove that R[a]/ (I) × (R/I)[x].

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 ø : R  → R’  be a ring homomorphism and let N be an ideal of R.  Let N’ be an ideal either of ø…

A: Ideal: A non-empty subset I of a ring R is said to be ideal in a ring I if it satiesfies following…

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Q: Let R be a commutative ring that does not have a unity. For a fixed a e R prove that the set (a) =…

A: Let R be a commutative ring that does not have a unity. For a fixeda∈ℝ, we need to prove that :…

Q: (B) 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: If S is a subring of a ring R, then S[a] is a subring of R[x]. Exercise 2.35.1 Prove this assertion!…

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Q: Let ? be a commutative ring with 1 and ? be a proper ideal of ?. Prove that ? is prime if  and only…

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Q: Prove that if (I,+,.) is an ideal of the Ring (R,+,.) then rad I=I ∩ rad R

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Q: Let T be ring containing elements e, f, both # 0T, such that e + f = 1r , e² = e, f² = f , and e · f…

A: We are given e+f = 1T,e2=e,f2=f and e.f=0T where T is the rings containing elements e,f both not…

Q: consider the mapping from M2(z) into Z,prove or disprove that this is a ring homomorphism

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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 R be a ring and S be a subring of R with OS, OR being the zero elements in S, R respectively.…

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Q: a is a unit in a ring R with unity, then a is not a zero divisor

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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: For a fixed element a of a ring R, prove that the set {x ϵ R I  ax = O} is a subring of R.

A: Given :            The ring R and a fixed element 'a' of R.   To prove that the set x ∈ R | ax =…

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Q: For any element a in a ring R, define (a) to be the smallest ideal of R that contains a. If R is a…

A: An ideal is a non-empty sub set I of a ring R, such that

Q: If ø is a ring homomorphism from R to S. Then i. ii. Prove that (kero) is an ideal of S. Prove that…

A: Given φ is a ring homomorphism from R to S. To prove: φkerφ is an ideal of S. Given, φ: R→S is a…

Q: element a ∈ R, define the Annihilator of a denoted as Ann(a), as Ann(a) = {r ∈ R| r.a = 0} Show that…

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Q: What does the notation R* mean with R being a ring with unity?  Let R be a ring with a subring S:…

A: What does the notation R* mean with R being a ring with unity?  Let R be a ring with a subring S:…

Q: Let R be a ring with a subring S: Prove or disprove: If a ∈ R is a unit, and a ∈ S, then a is also a…

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Q: Let R be a ring. The center of R is the set {x E R | ax = xa for all a in R}. Prove that the center…

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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 I = {(a, 0) | a e Z}. Show that I is a prime ideal, but not a maximal ideal of the ring Z×Z.

A: Ideal of a ring

Q: Prove that if (I,+,) is an ideal of the ring (R,+, ), then rad I In rad R. %3D

A: The term radical is used when we think about ideals and when we talk about ideals definitely…

Q: Let R be a commutative unitary ring and let M be an R-module. For every rERlet rM = {rx; x E M} and…

A: The complete solutions are given below

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: Prove that if u is a unit in a ring R, then -u is a unit in R.

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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: 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: Let R be a commutative ring. Prove that the principal ideal generated by the element x E R[x] is a…

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Q: If I is an ideal of a ring R, prove that I[x] is an ideal of R[x].

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Q: Q2: Prove that the intersection of any two ideals of a ring R is also an ideal.

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Q: Let f : R→ R' be a ring homomorphism of commutative rings R and R'. Show that if the ideal P is a…

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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: Is the mapping from Z5 to Z30 given by x → 6x a ring homomorphism? Note that the image of the unity…

A: In the given question we have to tell that " Is f(x)=6x is a ring homomorphism from (Z5,⊕5,⊗5) to…

Q: Exhibit a commutative ring R and an element x E R such that Z CR and x is NOT prime but irreducible…

A: Take R = Z[i√5]  Clearly, R is commutative ring and Z ⊆ R  Also, 2,3 ∈ R are not prime but…

Q: Prove that if u is a unit in a ring R, then u is a unit in R. -

A: According to the given information, It is required to prove that if u is a unit in a ring R then -u…

Q: If U, V are ideals of a ring R, let U + V = {u+ v:u E U,v E V}. Prove that U +V is also an ideal.

A: We have to prove the conditions of ideal

Q: Let R be a commutative ring with unity and r ∈ R. Prove that if ⟨r⟩ = R, then r is a unit. Consider…

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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: Prove that if (I,+,.) is an ideal of the Ring (R,+,.) then rad I= In rad R ???

A: Solution :

Q: Given that (I,+,.) is an ideal of the ring (R,+,.) Show that : a- the ring (R/I,+,.) may have…

A: We have to show that by example

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: 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…