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

Q: Show that the union of a chain I1 ⊂ I2 ⊂ .... of ideals of a ring R isan ideal of R.

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Q: Let S be a ring. Determine whether S is commutative if it has the following property: whenever æy =…

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Q: Let R be a ring with unity which has exactly one maximal left ideal M. Show that the only…

A: Given: R is a ring with unity which has exactly one maximal left ideal M. To show: The only…

Q: As a ring, z is isormorphic to nZ for all n>1. True False

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Q: Let R be a commutative ring with 10. Prove that R is a field if and only if 0 is a maximal ideal.

A: If R is a field, then prove that {0} is a maximal ideal. Suppose that R is a field and let I be a…

Q: Let T = Z - 5Z. Show that ZT is a local ring. What is its unique maximal ideal?

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Q: Let R be a ring of all real numbers , Show that H= { m+nV2 | m, ň E Z} is a subring of R.

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

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Q: Let R be a commutative ring. If I and P are ideals of R with P prime such that I ¢ P, prove that the…

A: The ideal quotient of P and I is P:I=x∈R : xI⊂P which is again an ideal of R. Given that P is a…

Q: Show that the centre of a ring R is a sub ring of R. And also show that the centre of a division…

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Q: Suppose that (R, +, . ) is a commutative ring with identity and it has nontrivial ideals, then R…

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

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Q: Let R be a ring such that a² = a for all a E R, and assume that R has an identity. Show that the…

A: Given R be a ring with unity 1 such that a2=a, for all a∈R. Let b∈R be a unit in R. Therefore b-1…

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

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

Q: The cancellation laws for multiplication are satisfied in a ring T F R, if R has zero divisor.

A: Here, given that The ring R with the cancellation law for multiplication holds in R. Let a,b,c∈R if…

Q: Any ring R is a Jacobsen radical ring, if it is a simple ring. O مبا O

A: JACOBSON RADICAL:- The radical of the base ring R is called its Jacobson radical and denoted by…

Q: Let R be a ring with unity. Show that (a) = { £ xay: x, y e R }. finite

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Q: Show that the centre of a ring R is a sub ring of R. And also show that the centre of a division…

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Q: Show that the set 4Z U 5Z is not a subring of the integer ring Z:

A: Z = { ....., -3, -2, -1, 0, 1, 2, 3, ..... }. 4Z = { ...., -8, -4, 0, 4, 8, .... }. 5Z = { ....,…

Q: 4 If R is a commutativ ring, show that the characteristic of R[X] is the same the characteristic of…

A: Given

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: Let R be a ring with 1. Show that R[z]/ (x) ~ R.

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

A: Yes it is true .

Q: Let S be a ring. Determine whether S is commutative if it has the following property: whenever ry =…

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

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

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Q: * If K is a simple ring, then K a Jacobsen radical ring. T F

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Q: Let R be a commutative unitary ring and let M be an R-module. For every r ERlet rM = {rx; x E M} and…

A: We shall solve first question only as you have asked more than one different question as per company…

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

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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 commutative unitary ring and let M be an R-module. For every rERlet rM = {rx; x E M} and…

A: The complete solutions are given below

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

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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: 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: -: Define subring. Is the set S is a subring of the ring M %3D Il 2 x 2 matrices?

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Q: Let (R, +, .) be a nontrivial ring with * identity, prove that 1 0

A: It is given that (R,+, .) be a nontrivial ring with identity. Now we have to show that 1≠0. So, (R,…

Q: Let R be a commutative ring of characteristic 2. Prove that : (a+ b) = a² +b² = (a - b)? v a, be R.…

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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: The rings Z and 5Z are isomorphic.

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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: Find all ring homomorphisms from R to Z5.

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

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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: Determine whether T={(a, -a) | a EZ} is a subring of ZxZ-

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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: Let R be a ring and M be an R-module. Ther

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