Given that (I,+,.) is an ideal of the ring (R,+,.) Show that : a- the ring (R/I,+,.) may have divisors of zero, even though (R,+,.) does not have any. b- if (R,+,.) is a principal ideal ring ,then so is the quotient ring (R/I,+,.)

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: 12. Let (I,+,') be an ideal of the ring (R,+, ·). Prove that (I,+, ·) is a primary ideal if and only…

A:

Q: Let I be an ideals of a ring R and S C R: (i) Show that InS is an ideal of S. Is it also an ideal of…

A: I is an ideal of a ring R and S is subring of R. So if x∈I∩S ⇒x∈I and x∈S sx∈I     for some s∈S and…

Q: Given a ring R and a ∈ R. Let S = {ra | r ∈ R}. Construct an example which shows: (a) S need not…

A:

Q: Let R be a commutative ring with unity, I an arbitrary ideal in R and P1,P2,...,Pn prime ideals in…

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: 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: It is known that (ℤ,⋆,⊙) where ? ⋆ ? = ? + ? − 1 and ? ⊙ ? = ? + ? − ?? is a ring. Does it have…

A: Please check step 2 for the solution.

Q: Let ? be a commutative ring with 1 and ? be a proper ideal of ?. Prove that ? is prime if  and only…

A:

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

A: 3

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: Suppose that R is a commutative ring and |R| = 30. If I is an idealof R and |I| = 10, prove that I…

A:

Q: If A, B, C are ideals of a ring R such that BCA, prove that An (B + C) = B+(AnC) = (A B) + (AnC).

A:

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 False True

A:

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

A:

Q: 20. Suppose R is a commutative ring and [R]= 30. If I is an ideal of R and |I| = 10, prove that I is…

A:

Q: Let R be a ring with unity and assume p, q, r ∈ R∗ Find (pqr)-1and also prove that it’s the inverse…

A:

Q: 18. Let (R, ,) be a commutative ring with identity and let N denote the set of nilpotent clements of…

A: a). As given that N is nonempty as 0∈N.  an=0 (n= positive integer) Then r∈R ran=rnan=0 Let take a,…

Q: Given that I = {(a,b) E R | a and b are even numbers} is an ideal of a ring R = Z × Z, construct…

A:

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

A:

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

A:

Q: 24. Let (R, +,) be a commutative ring with identity and a ER be an idempotent which is different…

A: R, +,· is said to be commutative ring if Suppose R is a non empty set such that for any two elements…

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

A:

Q: 17. Let H and K be ideals of a ring R. Show that HNK is an ideal of R.

A:

Q: Given a ring R and a ER. Let S = {ra|r ER}. Construct an example which shows S need not be an ideal,…

A: Consider the ring with unity  R=M2×2ℝ of all 2×2 matrices of real entries with usual matrix addition…

Q: given an example for a ring R has ideal I such that : VI #I + IC VI

A: Note:- If you don't understand anything or want anything else in this problem then please resubmit…

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…

A:

Q: 7. If f is a homomorphism from the ring (R, -+, ) onto the ring (R', +', '), show the following: a)…

A: Given that f is a homomorphism from the ring (R,+,.) onto the ring (R',+',.') The objective is to…

Q: If A and B are ideals of a commutative ring R with unity and A + B= R, show that A N B = AB.

A: Given Data: A and B are ideal of a commutative ring R with unity And A + B = R   For A⋂B = AB, The…

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

A:

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: Let M and N be ideals of a ring R and let H = {m+n | m∈ M, n ∈ N} (a) Show that H is an ideal of R.…

A:

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: Let (l,+,.) be an ideal of the ring (R,+,.) . Prove that (l,+,.) is a primary ideal if and only if…

A: Given problem is : Let (l,+,.) be an ideal of the ring (R,+,.) . Prove that (l,+,.) is a primary…

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

A:

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

A:

Q: if A, B, C are ideals of a ring R such that BCA, prove that An (B + C) = B+ (AnC) = (A B) + (AnC).

A:

Q: Consider the ring (Q,Ð,0), where the "addition" O and "multiplication" ® are defined as follows:…

A:

Q: If A, B, C are ideals of a ring R such that BCA, prove that An (B + C) = B+ (AnC) = (An B) + (AnC).

A:

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: @イイ全 | * (4 In the ring (Z, +,), an ideal (8Z, +,)is prime. True False * (5 The ring (Z30/(10),0.0)…

A: As per the guidelines I am supposed to do the first question among multiple questions. So I am…

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

A:

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: 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

A:

Q: An element a of a ring R is called nilpotent if an = 0 for some positive integer n. Show that the…

A: Given :            R is a commutative ring.            An element a∈R of a ring R is called…

Q: 17. Given that (I, 1.) in an ideal of the ring (R,+,), show that a) whenever (R,1,) is commutative…

A: Definition Two-sided Ideal of a ring : An ideal I of a ring R is called a two-sided ideal if it…

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

A:

Q: Let R be the ring Z[√−5]. (a) Prove that I = (2, 1 + √−5), the ideal generated by those two…

A: Hello. Since your question has multiple parts, we will solve the first part for you. If you want…