10. Let R be a commutative ring. The element a E R is called nilpotent if a' =U Tö SOIme positive integer n. Let I = {a €R: a is nilpotent}. Prove that I is an ideal of R.

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

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Q: Let be a commutative ring with unity of characteristic 3. Compute and simplify (a + b)° for all…

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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: Let M2(Z) be a ring and =(( ), a,b e Z}. Then S is ideal %D .of M2(Z) l choose one true Error

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Q: (3) The ring (4Z, +,.) has the following prime ideal ... (a) ((0), +,.) (b) ((8), +,.) (c) ((12),…

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

A: Subring Test : A non empty subset S of a ring R is a subring if S is closed under subtraction and…

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 Z,2, has exactly-- --maximal ideals O 2 3.

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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: If R is a commutative ring with unity and a e R, then (a) = {ar : reR}=aR.

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

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Q: If ne Z not prime, then the ring (Zn, +ni'n) is a) Integral domain b) field c) Division ring d) not…

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Q: O) be the smallest ideal of R that contains a. I6 Ris a commulative Ring uith unity , show That…

A: Given, a be the smallest ideal of R that contain a.If R is a commutative Ring with unity,   To prove…

Q: The ring Z3[i] has no proper ideals aya Math ele haw

A: O have proved the general result for arbitrary field.

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

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Q: 9. Suppose that (R,+, .) be a commutative ring with identity and x E rad R, then ..... .... (a) (x)…

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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: 9. Suppose that (R,+, .) be a commutative ring with identity and xE rad R, then (a) (x) = R (b) 1-x…

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Q: Let R be a ring with m elements. Show that the characteristic of Rdivides m.

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

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

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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: (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: (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 E = {a + bi : a, b ∈ Z, b is even}. 1. A subring S of a ring R is called an ideal of R if sr,…

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Q: 35. Show that the first ring is not isomorphic to the second. (a) Eand Z © Z × Zu and Z () ZXZ, and…

A: The objective is to show that the first ring of the following is not isomorphic to the second:

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

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

A: Prime Ideal:- An ideal P of a commutative ring R is prime if it has the following two properties:…

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: 9. Suppose that (R,+,.) be a commutative ring with identity and x E rad R, then (a) (x) = R (b) 1- x…

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

A: I jave used the definition of subring

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

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

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Q: (3) The ring (4Z, +,.) has the following prime ideal.. (a) ((0), +, .) (b) ((8), +,.) (c) ((12),…

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Q: The number of nilpotent elements in the ring Z23 is:

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

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

Q: Suppose I, J be ideals of a commutative ring R. Prove that IJ cIn).

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Q: The ring Zpg?, has exactly-------------maximal ideals O 2

A: 3

Q: Suppose R is a commutative ring with 1R# 0R. Show that if f (x) = ao + a1a + a2a ++a,n" is a unit in…

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Q: The ring (4Z, +,.) has the following prime ideal ... (a) ((0), +, .) (b) ((8),+,.) (c) ((12), +,.)…

A: Prime ideal sometimes behaves like a prime numbers . Let's firstly define prime ideal.

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

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Q: Show that the rings (3Z/60Z)/(12Z/60Z) and 3Z/12Z are isomorphic. Then show tha both isomorphic to…

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