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 ring Z[i] of Gaussian integers.

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: IF J is nil left ideal in an Artinian ring R, then J is

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

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Q: Consider the ring ℤ12. List the elements of the following principal ideals: (i) ⟨4⟩  (ii) ⟨9⟩…

A: Given ring is ℤ12.

Q: if A and B are ideals in a ring R such that A intersect B ={0}, prove that for every a in A and b in…

A: Let A and B are ideals of a ring R such that A∩B=0

Q: Let R be a commutative ring that does not have a unity. For a fixed a e R prove that the set (a) =…

A: Let R be a commutative ring that does not have a unity. For a fixeda∈ℝ, we need to prove that :…

Q: 12. Prove that the rings Z,[x]/(x² + x + 2) and Z,[x]/(x² + 2x + 2) are isomorphic.

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Q: 1. Let I and J be ideals of a ring R. Prove that IJ is an ideal of R.

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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: 1. Suppose that (R,+,.) is a ring and I is not maximal ideal in R. Then... ..... .... (a) 1 = R (b)…

A: {R, +, .) is a ring and it is given that I is not a maximal ideal in R.From this, we conclude that I…

Q: (B) Let I be a maximal proper ideal of commutative ring with identity R. Prove that R/I is a field.

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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: (b) Show that any nonzero element of the ring QIV2 = {a + bv2 | a, b e Q} is invertible, that is,…

A: b)  We have given that ,  ℚ2 = a + b2 / a , b ∈ ℚ We need to show that , for any non-zero element of…

Q: Prove that a ring with unity R has a unique maximal left ideal M if and only if R\M is the set of…

A: Prove that a ring with unity R has a unique maximal left ideal M if and only if R\M is the set of…

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: Prove that if (I,+,.) is an ideal of the Ring (R,+,.) then rad I=I ∩ rad R

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

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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 R be a ring and S be a subring of R with OS, OR being the zero elements in S, R respectively.…

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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: (5) I is maximal ideal of ring R if and only if R = (a, I) for any..... a in R O a in I a in R-{I} o…

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

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Q: Let R be a commutative ring with unity, and let c ER be a fixed element. (a) Prove that the set A =…

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Q: 1. Suppose that (R,+,.) is a ring and I is not maximal ideal in R. Then .... ..... (а) 1 3D R (b) 3J…

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Q: b) Prove that, if S is a ring with characteristic 0, then S infinite.

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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: 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 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: 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 commutative ring. Prove that the principal ideal generated by the element x E R[x] is a…

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Q: If I is an ideal of a ring R, prove that I[x] is an ideal of R[x].

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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: 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: 2- An example of two ideals A and B of the ring (Z12. +2-u) such thata B are: (a) A and B (c) A- and…

A: First we find all subgroup of Z12

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

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

A: Let R be a commutative ring with unity. If I is a prime ideal of R we have to prove that I[x] is a…

Q: a) If U and V are ideals of a ring R and let UV be the set of all those elements which can be…

A: We are given that U and V are the ideals of a ring R. Now, we need to show that UV is an ideal of R.…

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

A: R is ring with unity and I is ideal of R.

Q: 4: prove that Let R be a commutative ring with identity and I be a maximal ideal of R. Then the…

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Q: a. Let R and S be commutative rings with unities and f:R -S be an epimorphism of rings. Prove that S…

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Q: Prove that if (I,+,.) is an ideal of the Ring (R,+,.) then rad I= In rad R ???

A: Solution :

Q: Theorem 15.2.10 Let R be a principal ideal ring and a, b e R not both zero. Then a and b have a gcd…

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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: Find all values of a in Z5 such that the quotient ring Z,[x]/(p(x)) where p(x) = x³ + x² + ax + 4 is…

A: Solve the following

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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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…