2) Let M be a noetherian R-module over commutative ring R. If f : M M is onto ho- momorphism , show that f is one to one homomorphism.

Q: Construct a homomorphism of rings p:Z[i] → Z,

A: Consider the rings ℤi and ℤ2. Define a map φ:ℤi→ℤ by φa+ib=0 ∀ a,b∈ℤ. Let a+ib, c+id∈ℤi.…

Q: Let R be a commutative ring. Then R is an integral domain if and only if ab= ac implies that b = c,…

A:

Q: Let R be a ring with a finite number n of elements. Show that the characteristic of R divides n.

A: Given :  R is a ring with n elements. To prove :  The characteristic of a ring R divides the number…

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

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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: Prove that the only homomorphisms from Z to Z (Z being the ring of integers) are the identity and…

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

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

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Q: If a ring R has characteristic zero, then R must have an infinite number of elements. true or false

A: We have find that the given statement  "If a ring R has characteristic zero, then R must have an…

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

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Q: If fis a ring homomorphism from Zm to Z, such that f(1)= b, then b**2 = b*. True False

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Q: 5. If S is a subring of a commutative ring R, then R is an S-module with scalar multipli- cation…

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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 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 1={0,2} CZ. Prove if ring Z,/l is isomorphic with a ring Z,

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Q: If Ø: R → S is a ring homomorphism. The Ø preserves: All of these Nilpotent elements Units O…

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Q: Let R be a commutative ring with identity and I be ideal of R. Then I is primary if and only if…

A: The statement is true.

Q: 3) Given a commutative ring with unity 1 in R; where R is a ring with two maximal ideals M₁ and Show…

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Q: Show that Ø is a ring homomorphism.

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Q: If Ø: R→ S is a ring isomorphism. The Ø preserves: O All of these O Nilpotent elements O Idempotent…

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Q: If Ø: R → S is a ring isomorphism. The Ø preserves: O Nilpotent elements Units Idempotent elements O…

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

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Q: Prove that in a ring R having exactly one maximal ideal M, the only idempotents are 0 and 1.

A: Assume R is local and let M=R∖Rx. By assumption M is an ideal. It is also maximal because any ideal…

Q: Suppose that : R → S is a ring isomorphism. Then g-l is a ring homo- morphism, and hence a ring…

A: According to the given information:

Q: The set of all units of the ring Zg is

A: SOLUTION: The set of all units of the ring Z8={0,2,4,6} because, f(0)=f(2)=f(4)=f(6)=0

Q: Let R be a commutative ring with identity. Using the homomorphism theorem (Theorem 16.45) and…

A: Recall that in a ring A  not necessarily commutative and with an identity, an ideal M⊂A is a maximal…

Q: If f is a ring homomorphism from Z„ to Z„ such that f (1) = b, then b4*+2 = b*. O True False

A: Given f : Zm→Zn is a ring homomorphism such that f1=b Here, we have to check whether b4k+2=bk is…

Q: Let R be a finite ring and α ∈ R with α ≠ 0. If α is not a zero divisor, then α is a unit.

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

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Q: Let R be a commutative ring. Show that R[x] has a subring isomorphicto R.

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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 Ø is a ring homomorphism from Z, to itself of the form Øx) = ax,then show %3D that a2…

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Q: Let R be a ring with identity. If ab and a are units in R, prove that b is a unit.

A: Since you have asked multiple questions so as per guidelines we will solve the first question for…

Q: Let A = {" :a, b e Z} and let P: A → Z such that P(z) = a – b for every z e A. Prove that O is a…

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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: Prove that the only homomorphisms from Z to Z (Z being the ring of integers) are the identity and…

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Q: Find all ring homomorphisms from R to Z5.

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Q: Given a commutative ring R considered as a module over itself, determine all the module…

A: Given: A commutative ring R is considered as a module over itself.   To determine: All the module…

Q: Let A, B,C be rings. Let & be a ring homomorphism from A into B and ß be a ring homomorphism from B…

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

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Q: Let f: MM' be an R-module homomorphism, then

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

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Q: 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: 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: Theorem 12. Let R is a commutative ring and r e R,fe Hom (M, M'), then rfe. R M, M') defined by (rf)…

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Q: 3) Let R be a commutative ring with 1. Let A and B be two distinct maximal ideals of R. Show that AB…

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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: If Ø: R → S is a ring isomorphism. The Ø preserves: Units Idempotent elements

A:

Q: If Ø: R → S is a ring homomorphism. The Ø preserves: Units All of these Nilpotent elements…

A: The units in a ring are those elements which have an inverse under multiplication. An element x of…