Let R be a commutative ring with 1 and I be a proper ideal of R. Prove that I is prime if and only if R/I is an integral domain.
Q: Show that the polynomial ring Z4 [x] over the ring Z₁ is infinite, but Z₁ [x] is of finite…
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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 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: Give an example of a subring of a ring, say A, that is not an ideal of A.
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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 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: 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: Let R be a ring with unity which has exactly one maximal left ideal M. Show that the only…
A: Given: R is a ring with unity which has exactly one maximal left ideal M. To show: The only…
Q: The ring Zs(i) has no proper ideals True False
A: Given statement is false. Justification is given in step two.
Q: Let R be a commutative ring with 10. Prove that R is a field if and only if 0 is a maximal ideal.
A: If R is a field, then prove that {0} is a maximal ideal. Suppose that R is a field and let I be a…
Q: Let R be a commutative ring. Prove that HomR(R, M) and M are isomorphic R-modules
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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: 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 commutative ring with unity and let M be a maximal ideal of R such that M2 = {0}. Show…
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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: Suppose that (R, +, . ) is a commutative ring with identity and it has nontrivial ideals, then R…
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Q: Let R be a ring. If the only ideals of R are {0} and R itself, then R is a field.
A: We know the definition of , ideals of ring R. A non empty subset I of R is said to be an ideal of R…
Q: Suppose that R is a commutative ring and |R| = 30. If I is an idealof R and |I| = 10, prove that I…
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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: Let R be a commutative ring with identity and I be ideal of R. Then I is primary if and only if…
A: The statement is true.
Q: Let R be an integral domain. Prove that {0R} is a prime ideal. Let R be a ring and let p ∈ R be…
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Q: Let: ϕ:R → S be a ring homomorphism. Show that if ϕ is the overlying and M⊆R is maximal ideal, then…
A: Given that ϕ:R → S be a ring homomorphism. This implies that R and S are commutative rings with 1.…
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: Prove that in a ring R having exactly one maximal ideal M, the only idempotents are 0 and 1.
A: Assume R is local and let M=R∖Rx. By assumption M is an ideal. It is also maximal because any ideal…
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: 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 I be the set of all elements of a ring R that have finite additive order. Prove that I is an…
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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: Let R be a commutative ring. Show that R[x] has a subring isomorphicto R.
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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: (a) Let R be a commutative ring with M being maximal ideal in R then R/M is a field.
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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…
A: Given R is a commutative ring with identity 1R J is a proper ideal of R R/J…
Q: Let: ϕ:R → S be a ring homomorphism. Show that if ϕ is the covering and M⊆R is maximal ideal, then…
A: Given φ:R→S be a ring homomorphism. Let,, φ is covering and M⊂R is maximal ideal. To prove that…
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 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: a. Let R and S be commutative rings with unities and f: R → S be an epimorphism of rings. Prove that…
A: a) Let R and S be commutative rings with unities and f:R→S be epimorphism of rings. Let 0S and 0R…
Q: Show that a commutative ring R with unity is a field if and only if its only two ideals are (0) and…
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Q: If R1 and R2 are subrings of the ring R, prove that R1 n R2 is a subring of R.
A: R1 and R2 are subrings of the ring R, prove that R1∩R2 is a subring of R
Q: Let R be a commutative ring with unity and let M be a maximal ideal of R such that M2 = {0}. Show…
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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: Let R be a commutative ring with unity and let M be a maximal ideal of R such that M2 = {0}. Show…
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Q: Let M be a commutative ring with identity x and R be a maximal ideal of M. Then prove that R is…
A: Prime Ideal : A prime ideal S of a commutative ring R is a proper ideal of R such that a, b∈R and…
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: The ring Zs[i] has no proper ideals True False O O
A: We check whether Z8[I] has proper ideal.
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: 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 commutative ring with 1 ≠ 0. Prove that R is a field if and only if 0 is a maximal ideal.
A: We are given that R be a commutative ring with unity. We have to show that R is a field if and 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…
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- 15. Prove that if is an ideal in a commutative ring with unity, then is an ideal in .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.
- 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 .)If R is a finite commutative ring with unity, prove that every prime ideal of R is a maximal ideal of R.Exercises Let be an ideal of a ring , and let be a subring of . Prove that is an ideal of
- Let 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. )36. Suppose that is a commutative ring with unity and that is an ideal of . Prove that the set of all such that for some positive integer is an ideal of .21. Prove that if a ring has a finite number of elements, then the characteristic of is a positive integer.