1. Suppose that (R,+,.) is a ring and I is not maximal ideal in R. Then .... ..... (а) 1 3D R (b) 3J ideal in R such that I CJCR (c) (a) and (b) (d) either (a) or (b)

Q: Question 10

A: The given ideal is I=a,0/a∈ℤ.To prove that:I is a prime ideal, but not a maximal ideal of the ring…

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

A:

Q: 7. Suppose that (R,+,.) be a commutative ring with identity and (I,+,.) be an ideal of R. If I is…

A: Given that (R, +, .) be a commutative ring with identity and (I, +, .)  be an ideal of R such that I…

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: 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.

A:

Q: 4) Define on the cartesian product of sets R:=R x R = {(^, z) | A, z E R } an addition and a…

A: We first show that R is a commutative ring. That is we show that R is commutative group under…

Q: Show that N= (3x, y)|x, y ∈ Z  is a maximal ideal ofZ×Z. Note that you must show that it is indeed…

A:

Q: The ring Zpq²r has exactly-----------maximal ideals 1 2 3

A: 3

Q: (c) Show that the ideal generated by x² + y² + z² € C[x, y, z] is a prime ideal.

A: If <x2+y2+z2>is not a prime ideal, there should exist g,h∈ℝx, y, z s.t. x2+y2+z2 | gh,…

Q: Let K, I and J be ideals of ring R such that both I and J are subsets of K with I C J. Then show…

A: Given:- K,I and J be ideals of Ring R Such that I⊂J⊂K (i) Claim: K/I is subring of R/I ∀r.s∈R K⊂R…

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…

A:

Q: 1. Consider the ring Z[r]. Prove that the ideal (2, x) = {2f(x)+rg(x) : f(x), g(x) E Z[r]} is not a…

A:

Q: а. Let I = {a + bi: a, b E Z[i]: 3 divides both a and b} . Prove that I is a maximal ideal of the…

A:

Q: 3. Explain why the polynomial rings R[r] and C[r] are not isomorphic.

A: This is a problem of Abstract Algebra.

Q: 16. If R is a field, show that the only two ideals of R are {0} and R itself.

A:

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…

A:

Q: Show that if R is a ring with unity and N is an ideal of R such that N R, then R/N is a ring with…

A: Since we have given that R is a ring with unity and N is a proper  ideal of R . We prove R/N is a…

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

A:

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…

A:

Q: Let I = {a + bi: a, b e Z[i]: 3 divides both a and b}. Prove that I is a maximal ideal of the ring…

A: Given that  I=a+ib|a,b∈ℤ and 3 divides both a and b To prove I is maximal ideal of ℤi. To prove this…

Q: The ring Z3[i] has no proper ideals O True O False

A: Yes it is true .

Q: Show that for any nontrivial ideal I of Z[i], Z[i]/I is finite.

A:

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

A:

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

A:

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

A:

Q: 7. The number of distinct homomorphisms from the ring (Z, +,.) onto the ring (Ze, +,.) is (a) 0 (b)…

A: Number of distinct onto homomorphism is

Q: Consider the ring of polynomials Q(z) , x²-1∈Q(z) Is aprinciple ideal ? Is a maximal ideal?

A:

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

A: Let R be a ring. A two-sided ideal I of R is called maximal if I ≠ R and no proper ideal of R…

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: (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: Let R be a ring with unity 1, and S = {n.1 : n E Z} . Then S'is Ra subring of Rnot a subring of

A: Let, x ,y in S. So, x = n•1 and, y = m•1 for some n, m in Z.

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

A: The given statement is ,The ring Z8i has no proper ideals.

Q: Prove that R[x2, x3] is a subring of the ring R[x].

A:

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

A:

Q: Let R be a ring with unity 1, and S = {n.1 : n E Z}.Then S'is Ra subring of Rnot a subring of

A: A non-empty set R with two binary operations addition(+) and multiplication(·) is called ring if it…

Q: (a) Let S {C ): a, 6, c e Z, where 0 denotes the usual integer zero. Given that S is a ring under…

A:

Q: Let R be a ring and S a subring of R. Prove that if 1R ∈S, then S has unity and 1S =1R

A:

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: Show that if R is a ring with unity and N is an ideal of R such that N ≠ R , then R / N is a ring…

A:

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

A:

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

A: We check whether Z8[I] has proper ideal.

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: In Z[x], let I = { f(x) E Z[x]: f(0) is an even integer). 1) Prove that I is an ideal of Z[x]. 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: 1. Suppose that p: R → S is a ring homomorphism and that the image of o is not {0}. If R has a unity…

A:

Q: Theorem 2.7. IfI, I, I2, . .., I, are ideals in a commutative ring R, then: (i) Rad (Rad I) Rad I;…

A: The solution for the above question is as shown below.