The ring Z75 has exactly-----------maximal ideals 4 O 2 6 3
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: Let R be a ring with unity 1 and char (R) = 4. %3D Then R contains a subring isomorphic to Q ZO Z3 O
A: IN the given question, Given that: R is a ring with unity 1 and char(R)=4. we have to find: we have…
Q: 3. Let R be a commutative ring with unity. The intersection of all maximal ideals of R is called the…
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Q: 4. The ring (Z, +,.) the ideal (12) containing in the following maximal ideal ... (a) (4) (b) (5)…
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Q: There are.... Polynomials of degree atmost n in the polynomial ring Z, (x O none O5+5^n O 5^(n+1) O…
A: The general form of the polynomial of degree n is Pn(x)= a0+a1x+a2x2+...+anxn .
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: 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 1. Show that S = {n· 1 | nE Z} is a sub- ring of R.
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Q: The ring 5Z is isomorphic to the ring 6Z OTrue O False
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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: Find all pro pes ideals of the ring (Z24s +24 '24)·
A: All Ideals of z_24 are 8..... And proper ideals are 6
Q: There are.... Polynomials of degree atmost n in the polynomial ring Z, [x]. 5^n O 5+ 5^n Onone…
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Q: The cancellation laws for multiplication are satisfied in a ring T F R, if R has zero divisor.
A: Here, given that The ring R with the cancellation law for multiplication holds in R. Let a,b,c∈R if…
Q: There are . Polynomials of degree atmost n in the polynomial ring Zs (x]. O 5 + 5^n O 5^n O 5^(n+1)…
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Q: 16. Let R be a commutative ring with unity and let N= {a e R| a" = 0 for some n e Z*; n2 1} Show…
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Q: a) The idempotents Of (Z6,0,0,) are ONLY 0, b) The number 161 is an irreducible element in Z[i] c) A…
A: As per the company rule, we are supposed to solve the first three sub-parts of a multi-parts…
Q: The ring Z is isomorphic to the ring 3Z O True False
A: Solution:
Q: Show that the set 4Z U 5Z is not a subring of the integer ring Z:
A: Z = { ....., -3, -2, -1, 0, 1, 2, 3, ..... }. 4Z = { ...., -8, -4, 0, 4, 8, .... }. 5Z = { ....,…
Q: Let R be a commutative ring with unity and let N= {a e R| a" = 0 for some n e Z"; n> 1} Show that N…
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Q: (b) If M is a maximal ideal of a ring R then M is a prime ideal of R.
A: Given: If M is a maximal ideal of a ring R, then M is a prime ideal of R To prove or disprove the…
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: (4) Let Z12 be a ring of integer modulo 12. Then there are.....maximal ideals of Z12- O (i) 4 O (ii)…
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Q: Consider the ring of polynomials Q(z) , x²-1∈Q(z) Is aprinciple ideal ? Is a maximal ideal?
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Q: 17. Let H and K be ideals of a ring R. Show that HNK is an ideal of R.
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Q: Let R be a ring with unity 1. If 1 has infinite order under addition then the characteristic of R is…
A: The order of an element a in a ring R is the least positive integer n such that n·a=0. If no such…
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: There are... Polynomials of degree atmost n in the polynomial ring Z, [x]. 5an 5+5An 5 (n+1) none
A: Given :- To find :- the number of Polynomials of degree atmost n in the polynomial ring Z5[x] .
Q: (10) Let I = (4) be a principal ideal of integer %3| ring Z, Then I is. . ideal * Primary Prime O…
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Q: 30. Let R be a ring with identity lr and S a subring of R with identity 1s. Prove or disprove that…
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Q: The ring 5Z is isomorphic to the ring 6Z True O False
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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: 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: 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: Let be a commutative ring with unity of characteristic 3. Compute and simplify (a + b)6 for all…
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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: 5. Let A and B be two ideals of a commutative ring R ith unity such that A + B=R. Show that AB =…
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Q: Q3: Prove that the ring of rational numbers (Q, +,.) is division ring
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Q: 8. List all ideals of the ring Z12-
A: Ideals of the ring Z12
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 ring Zpg?, has exactly-------------maximal ideals O 2
A: 3
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: If I1 and I2 are two ideals of the ring R, prove that Ii n 11 ∩ I 2 is an ideal of R.
A: Given I1 and I2 are two ideals of the ring R To prove : I1∩I2 is an ideal of R.
Q: 2) Let P + Q be maximal ideals in a ring R and a,b elements of R. Show that there exists c E R, such…
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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…
Q: The ring 3z is isomorphic to the ring 5z True False
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Q: There are .. Polynomials of degree atmost n in the polynomial ring Z,[x O 7^n O 7 + 7^n O 7^(n+1) O…
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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…
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- 8. Prove that the characteristic of a field is either 0 or a prime.15. In a commutative ring of characteristic 2, prove that the idempotent elements form a subring of .11. a. Give an example of a ring of characteristic 4, and elements in such that b. Give an example of a noncommutative ring with characteristic 4, and elements in such that .
- If R is a finite commutative ring with unity, prove that every prime ideal of R is a maximal ideal of R.33. An element of a ring is called nilpotent if for some positive integer . Show that the set of all nilpotent elements in a commutative ring forms an ideal of . (This ideal is called the radical of .). a. Let, and . Show that and are only ideals of and hence is a maximal ideal. b. Show that is not a field. Hence Theorem is not true if the condition that is commutative is removed. Theorem 6.22 Quotient Rings That are Fields. Let be a commutative ring with unity, and let be an ideal of . Then is a field if and only if is a maximal ideal of .