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

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: 4. The ring (Z, +,.) the ideal (12) containing in the following maximal ideal ... (a) (4) (b) (5)…

A:

Q: (4) The ring (Z, +,.) has the following not maximal ideal (a) ((11), +,.) (b) ((31), +, .) (c) ((0),…

A: The given question is solved with explanation below.

Q: Find

A: We know that,  All ideals of Zn are kZn, Where k is divisor of n.

Q: Let R be a ring with a finite number n of elements. Show that the characteristic of R divides n.

A: Given :  R is a ring with n elements. To prove :  The characteristic of a ring R divides the number…

Q: The ring Z,2, has exactly-- --maximal ideals O 2 3.

A:

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: Find all maximal and prime ideals of Z10-

A:

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: prove The intersection of two maximal ideals is maximal.

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Q: what are the prime ideals of Z2xZ2

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Q: Let R be a ring with unity. Show that {а)3D( 2 хау: х, у є R}. finite

A:

Q: What are the maximal ideals in Z12? , O , O , O

A: Solution: Definition: A maximal ideal of a ring R is an ideal M≠R such that there is no proper ideal…

Q: is a maximal ideal of Qlz).

A:

Q: 2. Define reducible and irseducible elements ring R and irreducible elements in Zu, Zq nand Zio R…

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Q: Show that the nilpotent elements of a commutative ring form a subring

A: We have to prove that the nilpotent elements of a commutative ring form a subring. In order to do…

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

A: Given

Q: The ring (Za. +3i3) can be imbedding in the ring (Z12-+12'12). True False

A:

Q: In the ring Z Z , I = {(0, b)|b € Z} is : maximal ideal prime not maximal neither prime nor maximal

A:

Q: 30. Show that in a PID, every proper ideal is contained in a maximal ideal. [Hint: Use Lemma 45.10.|…

A: To prove -  In a Principal ideal domain(PID), every proper ideal is contained in a maximal ideal.

Q: The ring Z75 has exactly-----------maximal ideals 4 O 2 6 3

A:

Q: 4 IF Risa Commutativ ring, Show that the characteristic of REX] is the Same as the characteristicof…

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Q: Construct the adition and multiplication tables for the ring Z/4Z.

A: Given: A ring ℤ/4ℤ. To construct: Addition and multiplication tables for the ring ℤ/4ℤ.

Q: Let Z₁2 be a ring of integer modulo 12. Then there are.....maximal ideals of Z12. O (1) 4 O (ii) 2 O…

A: A detailed solution is given below

Q: Show that the ideal of (5) in the ring of integers Z is the maximal ideal.

A: An ideal A in a ring R is called maximal if A ≠ R and the only ideal strictly containing A is R. In…

Q: (4) Let Z12 be a ring of integer modulo 12. Then there are.....maximal ideals of Z12- O (i) 4 O (ii)…

A:

Q: Show that the subring of an artinain ring may not be artinian???

A: According to the given information, it is required to show that subring of an Artinian ring may not…

Q: Let K be integer ring module 12 and let I=([4]) and J-([6]) be ideals of K. Then ([0])

A: Let * be integer ring module 12.  Let  I=4J=6

Q: How many ring homomorphisms Z/35→Z/7  are there?

A: Given that ℤ35→ℤ7

Q: 2. In the ring (4Z, +,.), the ideal (8) is (a) not prime (b) maximal (c) maximal and not prime (d)…

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Q: 7) There is no proper non-trivial maximal ideals in (Z2₁, because is a maximal ideal in Z2₁, ,0 )…

A: Solution

Q: (17) Prove that the ring Zm Xx Z, is not isomorphic to Zmn if m and n are not relatively prime.

A: We have to prove given property:

Q: (5) In commutative ring with identity each proper ideal is contained in prime ideal. От F

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Q: Prove or disprove that Q(√3) and Q(√-3) are ring-isomorphic.

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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: The number of zero divisors of the ring Z4 O Z, is

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Q: It is known that 28= {0, 1, 2, 3, 4, 5, 6.73 is a Ring. H = { 0,43 is a subring of Z8. show that…

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Q: The cancellation laws for multiplication are satisfied in a ring R, if R has zero divisor.

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Q: (5) In commutative ring with identity each proper ideal is contained in prime ideal. От OF

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

A: Ideals of the ring Z12

Q: The ring Z,2, has exactly--- --maximal ideals 1 2.

A:

Q: Consider the ring of polynomia! Q [ Z] 2 -1 €Q[x] is aprincipl ideal 5

A: In the question it is asked to find whether <x2-1> is a principal ideal.

Q: Let R be a commutative ring with identity. Show that every maximal ideal is also a prime ideal. Give…

A: i just used the properties of maximal ideal to prove the result

Q: The ring (4Z, +,.) has the following prime ideal ... (a) ((0), +, .) (b) ((8),+,.) (c) ((12), +,.)…

A: Prime ideal sometimes behaves like a prime numbers . Let's firstly define prime ideal.

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

A:

Q: In the ring Z Z , I = {(0, b)|b € Z} is maximal ideal prime not maximal neither prime nor maximal

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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: In the ring Z4 O Z4,1 = {(0,b)|b € Z4} is: O neither prime nor maximal O prime not maximal maximal…

A: