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

Q: The ring Z pg?, has exactly------------maximal ideals 2 3 1 4

A: An ideal I in Zn  is maximal if and only if I=⟨p⟩ where p is a prime dividing n.

Q: In commutative ring with identity each proper ideal is contained in prime ideal. O T OF

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Q: Let (?,+, ⋅) be a ring with additive identity 0. Prove that for all x∈?, 0⋅x=0 and x⋅ 0 = 0.

A: We know that if (R,+,.) is a ring then (R,+) is an abelian group. And in abelian group, cancellation…

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…

Q: Verify in the following statement involving the ideal generated by (a1, a2 , ... , ak ) in the ring…

A: To show that 2,3=ℤ Let 2,3=2x+3y: x,y∈ℤ -------(1) gcd2,3=1 , Thus 2m+3n=1 for some…

Q: Let R be a ring with unity 1 and char (R) = 4. Then R contains a subring isomorphic to

A: Let R be a ring with unity 1 and char(R)=4.Then R contains a subring isomorphic to________

Q: The ring 5Z is isomorphic to the ring 6Z OTrue O False

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Q: やthe ring KLx,り,z3 wherc K is a field. Prove (x2 - (y1)) that is a K[x, り,そI . Prime ideal of

A: Given:- Ring  K[x ,y ,z]   where k is a field. To Prove:-  [x z-(y2+1)]

Q: Verify in the following statement involving the ideal generated by (a1, a2, ... , ak )) in the ring…

A: To prove that: (2, 5)=ℤ

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: (3) Let A be commutative ring with identity, then A has just trivial ideals iff A is ....... O…

A: Here you have posted multiple question, So as per the policy I can answer only first question for…

Q: Suppose that (R, +, . ) is a commutative ring with identity and it has nontrivial ideals, then R…

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Q: If A and B are ideals of a ring, show that the product of A and B, AB = {a,b, + a,b, +· · · + a,b,l…

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

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Q: Prove that every field is a principal ideal ring.

A: We’ll answer the first part of this question since due to complexity. Please submit the question…

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

A: Solution:

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: Is it true that if S is a unital subring of a unital ring, then the identity elements of the two…

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Q: 21. If R is a ring, prove that R[x] ~R, is the ideal generated by x.

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Q: 4 If R is a commutativ ring, show that the characteristic of R[X] is the same the characteristic of…

A: Given

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 every nE in the (Z , + , . ) ring, the I=nZ subring is an ideal. please show

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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 finite commutative ring with unity.  Show that every prime ideal in R is a maximal ideal.

A: We have to show that if R be a finite commutative ring with unity then every prime ideal in R is a…

Q: Let R = {0, 1, x}, where 0, 1, and x are distinct, and let (R, +, ⋅) be a ring with additive…

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

A: Given a mapping,

Q: IN denotes the set of noninvertible ele conditions are equivalent: (N,+,) is an ideal of (R,+, ), p)…

A: Given N denote the set of non-invertible  elements of R.

Q: Prove that the numbers 3 and 7 are indecomposable in the ring Z[V-5], but the number 5 is…

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Q: 2. Let R be a commutative ring with unity. If I is a prime ideal of R, prove that I [x] is a prime…

A: Given R be a commutative ring with unity and if I is a prime ideal of R. Then we have to prove that…

Q: Let (R,+, ⋅) be a ring with additive identity 0. Prove that for all x∈R, 0⋅x=0 and ? ⋅ 0 = 0.

A: Solution

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: The ring Zpg²r has exactly-- ---maximal ideals 2 1 4

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: The rings Z and 5Z are isomorphic.

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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: Let be a commutative ring with unity of characteristic 3. Compute and simplify (a + b)6 for all…

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Q: If R is a commutative ring, shw that the characteristic of R[x] is the same as the characteristic of…

A: Given: R is commutative ring

Q: Let R be a commutative ring with unity and let N={ aER | a"=0 for nez*, n>1}. Show that N is an…

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Q: The number of nilpotent elements in the ring Z23 is:

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Q: What are all of the distinct generators of Z17 with respect to multiplication? What are all the…

A: We’ll answer the first question since the exact one wasn’t specified. Please submit a new question…

Q: Suppose I, J be ideals of a commutative ring R. Prove that IJ cIn).

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Q: The ring Zpq?r has exactly------------maximal ideals O 3 2

A: Thanks for the question :)And your upvote will be really appreciable ;)

Q: The ring Z,3 has exactly------------maximal ideals 3 4 1 O O O O

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Q: Prove directly that a maximal ideal is irreducible.

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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 3z is isomorphic to the ring 5z True False

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Q: is union of two ideal rings R, an idea of R? prove or give counter example

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Q: Find the maximal ideals and the prime ideals in the ring Z6.

A: We have to find the maximal ideals and the prime ideals in the ring Z6

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: 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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