4. Let p: R S be a ring homomorphism. Show that J = ker p is a prime ideal if S is a domain. Show that J is a maximal ideal if S is a field and o is surjective.

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: D Let I be an ideal of ring R Such Hhut when cver R is Commutakive with idlentily then so is the…

A: Let R be a ring and let I be an ideal of R. We say that I is prime if whenever ab ∈ I then either 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: 6. Suppose that (M,+,.) be a maximal ideal of the commutative ring with identity (R, +,.) and x E M,…

A: Since, given that (M, + .) Is a maximal ideal of the commutative ring with identity (R, +, .)  and…

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

A:

Q: If Ø: R S is a ring homomorphism. The Ø preserves:

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: 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: Suppose that R and S are commutative rings with unities. Let ø be a ring homomorphism from R onto S…

A:

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

A:

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

A:

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

A:

Q: consider the mapping from M2(z) into Z,prove or disprove that this is a ring homomorphism

A:

Q: Let: ϕ:R → S be a ring homomorphism. Show that if ϕ is the overlying and M⊆R is maximal ideal, then…

A: Given that  ϕ:R → S be a ring homomorphism. This implies that R and S are commutative rings with 1.…

Q: (1) For every ring R and R-module M below, determine whether M 0 and prove your answer. (a) R= Z, M…

A: We evaluate elementary tensors and prove that they are 0.

Q: 5. Suppose that (R, +,.) is an infinite commutative ring and it has no nontrivial ideals, then R…

A: 5.R,+,.is an infinite commutative ring and it has no nontrivial ideals then R forms 6.suppose that…

Q: Q1: Let R be a commutative ring with Char(R) = 2 and let p:R → R be defined such that o (a) = a².…

A:

Q: If ø is a ring homomorphism from R to S. Then i. ii. Prove that (kero) is an ideal of S. Prove that…

A: Given φ is a ring homomorphism from R to S. To prove: φkerφ is an ideal of S. Given, φ: R→S is a…

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

A: Yes it is true .

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

A:

Q: Q::Let S1 and S2are two subrings of a ring (R, +,.), prove that S, U S2 is subring of R iff either…

A: 1 Let S1 and S2be two subrings of R,+,.. First suppose that either  S1⊆S2 or S2⊆S1 we will prove…

Q: Q1: Let S, and Szare two subrings of a ring (R, +,.), prove that S, USz is subring of R iff either…

A:

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: (a) Let R be a commutative ring with M being maximal ideal in R then R/M is a field.

A:

Q: 7. a) Prove that every field is a principal ideal ring. b) Consider the set of numbers R {a+ bV2|a,…

A:

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: Let: ϕ:R → S be a ring homomorphism. Show that if ϕ is the covering and M⊆R is maximal ideal, then…

A: Given φ:R→S be a ring homomorphism. Let,, φ is covering and M⊂R is maximal ideal.  To prove that…

Q: 7. a) Prove that every field is a principal ideal ring. b) Consider the set of numbers R = {a+…

A: a) Let F be a field. We know that field has no proper ideals. The ideals of F are 0 and F only. The…

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

A: a) Let R and S be commutative rings with unities and f:R→S be epimorphism of rings. Let 0S and 0R…

Q: Q: Let S, and Szare two subrings of a ring (R, +,.), prove that S, US2 is subring of R iff either S,…

A: By supposinɡ S1 and S2 as two subrinɡs of rinɡ (R, +, .) To prove that S1∪ S2 is subrinɡ of R if and…

Q: 1. let S: :1 xg*Z a) Show that is a ning. ning. b) Show that ¢: S Z olefined by $ ( [Š 87)-* is a…

A: Solution of part(a): It is given that S=x0y0, x,y∈ℤ. We know that the sum and product of two 2×2…

Q: Q2) Let(M₂ (R), +..) be a ring. Prove H = {(a) la, b, c = R}is a subring of (M₂ (R), +,.).

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: 18. Let p:C + C be an isomorphism of rings such that e(a) - a for each ae Q. Suppose r E Cisa root…

A: Given: Let ϕ:ℂ→ℂ be an isomorphism or rings such that ϕ(a)=a for each a∈ℚ. Suppose r∈ℂ is the root…

Q: R→s be a ring homomorphism. ve that if K is a subring of then R f(K) is a subring of S ve that c is…

A: Given, if f:R→S be a ring homomorphism. (i) To prove that if K is a subring of R then f(K) is a…

Q: 6. Suppose that (M,+,.) be a maximal ideal of the commutative ring with identity (R, +,.) and x E M,…

A:

Q: Let M be a proper ideal in a Boolean ring R with unity. Prove that (1) R/M is a Boolean ring and…

A:

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

A:

Q: Iffis a ring homomorphism from Zm to Zn such that f(1) = b, then b4k+2 = bk. True False

A:

Q: Let R be a commutative ring with unit element .if f(x) is a prime ideal of R[x] then show that R is…

A: Given R be a commutative ring with unit element. If f(x) is a prime ideal of R[x] then we have to…

Q: Iff is a ring homomorphism from Zm to Zn such that f (1)=b, then b*+2 = b*. False True

A:

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

A:

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: Let R be a commutative ring. Prove that HomR (R, M) and M are isomorphic R-modules.

A: “Since you have asked multiple questions, we will solve the first question for you. If you want any…

Q: Give the following theorem (without proof): If (R, +, ·) is a ring, and S C R then what is the…

A: That's easy. Thumb up. Have a great day!!!

Q: 2) Let M be a noetherian R-module over commutative ring R. If f : M M is onto ho- momorphism , show…

A: Please check step 2 and 3 for solution.