Show that the union of a chain I1 ⊂ I2 ⊂ .... of ideals of a ring R isan ideal of R.

Q: Let I be an ideal in a ring R with unity. Show that I = R if and only if I contains a unit.

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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: 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 1 and I be a proper ideal of R. Prove that I is prime if and only…

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Q: The ring Zs(i) has no proper ideals True False

A: Given statement is false. Justification is given in step two.

Q: 12. Prove that the rings Z,[x]/(x² + x + 2) and Z,[x]/(x² + 2x + 2) are isomorphic.

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Q: 1. Let I and J be ideals of a ring R. Prove that IJ is an ideal of R.

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Q: 1. Suppose that (R,+,.) is a ring and I is not maximal ideal in R. Then... ..... .... (a) 1 = R (b)…

A: {R, +, .) is a ring and it is given that I is not a maximal ideal in R.From this, we conclude that I…

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: (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: Let T = Z - 5Z. Show that ZT is a local ring. What is its unique maximal ideal?

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Q: 3. Prove that an ideal I in a ring R is the whole ring if and only if 1 e I.

A: Question: Prove that an ideal I in a ring R is the whole ring if and only if 1∈I. Proof: We have to…

Q: is the ring 2Z isomorphic to the ring 4Z??

A: We can apply definition

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. 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: Let Roll no be a ring with ideals I and J , such that I ⊆ J . Then J/I is an ideal of Roll no/I .

A: Ideal: A non-empty subset I of  a ring R is called ideal in a ring R if following conditions holds:…

Q: Let R be a ring with 1 0. Prove or disprove: (a) if R has no ideals other than {0} and R, then R is…

A: Given statement is false. Justification is in step 2

Q: Let 1={0,2} CZ. Prove if ring Z,/l is isomorphic with a ring Z,

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Q: The ring Z-[i] has no proper ideals True False

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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: 21. A ring is said to be a local ring if it has a unique maximal ideal. If (R,+,) is a local ring…

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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: 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: 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: a. Is the ring 2Z isomorphic to the ring 3Z?b. Is the ring 2Z isomorphic to the ring 4Z?

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Q: The ring Zs[i] has no proper ideals

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

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Q: 1. Suppose that (R,+,.) is a ring and I is not maximal ideal in R. Then .... ..... (а) 1 3D R (b) 3J…

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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: 7. a) Prove that every field is a principal ideal ring. b) Consider the set of numbers R {a+ bV2|a,…

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Q: A proper ideal J of R is a maximal ideal if and only if R/J is a field.

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

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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: a) If U and V are ideals of a ring R and let UV be the set of all those elements which can be…

A: We are given that U and V are the ideals of a ring R. Now, we need to show that UV is an ideal of R.…

Q: Let I= {(a, 0)| a eZ}. Show that I is a prime ideal, but not a maximal ideal of the ring Z×Z. Id if…

A: We knew that an ideal I of a ring R is said to be prime ideal if for a ,b ∈ R and ab ∈ I this imply…

Q: The ring Z,3 has exactly-------------maximal ideals

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Q: Find all ring homomorphisms from R to Z5.

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Q: '. Suppose that (R, +,.) be a commutative ring with identity and (I, +,.) be an ideal of R. If I is…

A: R is ring with unity and I is ideal of R.

Q: is the ring 2Z isomorphic to the ring 3Z?

A: No ring 2Z is not isomorphic to the ring 3Z

Q: 8. List all ideals of the ring Z12-

A: Ideals of the ring Z12

Q: Find all the principal ideals of the ring (Z6, O, 0 0 ).

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

A: Solution :

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: The ring Z-[i] has no proper ideals

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Q: 21. A ring is said to be a local ring if it has a unique maximal ideal. If (R,+, ) is a local ring…

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Q: The ring Zp2, has exactly-----------maximal ideals

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