Let M and N be R-modules. Prove that ifM is simple (Exercise 2.8) then every non- zero R-morphism f: M N is a monomorphism; and that if N is simple then every non-zero R-morphism f : M → N is an epimorphism.

Q: Let f:R→s be a ring homomorphism. (1) Prove that if K is a subring of R then f(K) is a subring of s.…

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Q: I - A Define an injective module and then prove that a module E is injective if and only if for each…

A: Given: Let us define an injective module and then show that a module E is injective if and only if…

Q: 5. Suppose that f: X →Y and g : Y → Z are bijections. Prove that gof is a bijection and that (go f)-…

A: We will find out the required solution.

Q: Prove that subset S = {a + bi ∈ Z[i] | b is even} is not an ideal

A: A ring R is a set with two binary operations, addition and multiplication, satisfying several…

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

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Q: prove this: The minimal sigma field contain C is unique

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Q: (Z × Z)/{(4, 7)) ~ Z

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Q: Let A e M2(Q). 2 Define a ring homomorphism ф: Q] —> М2(Q), f(x) → f(A). (i) Show that A² – 2A – 3 =…

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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: For any positive integer m, show that every subgroup of Zm is an ideal of Zm-

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Q: If E is an extension of F and f (x) e F[x] and if o is an automorphism of E leaving every element of…

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

A: Ring homorphism

Q: 4. Show that the collection C = {[a, b) | a < b, a and b rational} is a basis that generates a…

A: Given:         l = {[a,b]} | a&lt;b , a and b rational}

Q: is an R-module homomorphism snomorphism if and only if k

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Q: Show that there is no associative algebra A such that the Lie algebra arising from A is Glo(C)

A: The center of a lie algebra L is, C(L)={x∈L:[x,y]=0 for all y∈L} Since[x,y]=0 so x and y commutes.…

Q: - The mapping f: Z → Z defined by f (n) = 5n is a ring homomorphism. a) True b) False

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Q: Let E be the set of all positive even integers. Prove that E is countably infinite by defining a map…

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Q: Let M, and M2 be two R-modules and N, and N2 be two submodules of M, and M2 respectively. Also, let…

A: Here we need just one fact that if f:X→Y is a function defined from X to Y and if y∈Y, then f-1(y)∈X…

Q: (b) If M is a maximal ideal of a ring R then M is a prime ideal of R.

A: Given: If M is a maximal ideal of a ring R, then M is a prime ideal of R To prove or disprove the…

Q: 2. Let R be a ring: The center of R is the set 3XER: ax= xa vae R? Prove that the center of a ring…

A: Let R be a ring .We have to show that centre of ring is a subring of R

Q: , then A'(V) is a free K – module of rank

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Q: Determine whether the following mapping is a ring homomorphism h: Z[V2]→ Z[v2] defined by h(a +…

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Q: Let p: M M' be a homomorphism of R-modules and let N be an R-submodule of M. We define the image of…

A: I have used the submodule criterion and definitions of generator's of module and basis of module

Q: 10. By using the Fundamental Homomorphism Theorem, prove the following statements: (a) (Z x Z)/((4,…

A: We will solve all the four parts for you, using basic knowledge of group isomorphisms.

Q: Let R be a ring. The center of R is the set {x E R | ax = xa for all a in R}. Prove that the center…

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Q: (Z × Z × Z)/{(1, 3, 9)) ~ Z × Z

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Q: Recall that Q8 =. Prove that the map o from Qg to GL2(C) defined on generators by P(i) = |V-1 And…

A: A group homomorphism Φ : G→H is a map such that for any g1, g2 ∈G, Φg1g2=Φg1Φg2. The kernel of a…

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: Define which of the followings are ring homomorphisms from M2(Z) to Z. (:)- a (a) Ha (projection…

A: Ring homorphism

Q: - The mapping f:Z → Z defined by f (n) = 5n is a ring homomorphism. a) True b) False

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Q: Define an injective module and then prove that a module E is injective if and only if for each…

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Q: 16. Let f: R S be a ring homomorphism with J an ideal of S. Define I= {r ER| f(r) € J} and prove…

A: Given :  f : R →S be a ring homomorphism with J an ideal of S. To prove :  I = r∈R : f(r)∈J  is an…

Q: (Z × Z)/((5, 5)) ~ Zz x Z

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Q: Let f:R→s be a ring homomorphism. (1) Prove that if K is a subring of R then f(K) is a subring of s.…

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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 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: Prove that the mapping x --&gt; x6 from C* to C* is a homomorphism.What is the kernel?

A: Given: the mapping x --&gt; x6 from C* to C* is a homomorphism.

Q: Let MR be a finitely generated module over a ring R. Prove: (a) Every proper submodule of M is…

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Q: (Z × Z × Z)/((1, 3, 9)) ~ Z × Z

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Q: Indicate such a subring of the ring P[x], which contains P and is different from P, but is not…

A: Image is attached with detailed solution.

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.

Q: Show that if S is a ring and M is a free S-module on a countably infinite set e1, e2, .. ., then M…

A: It is given that

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: 6. Let F be a finite field. Prove that if f : F → F is a ring homomor- phism, then f is an…

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Q: 3) Let f: (A, SA) →(B,S.) be an order isomonphism and, Sa={xEA: * Kaaf#$ and Spa) = ly eB: Prove…

A: Consider the given information: Let f:A,≤A→B,≤B be an order isomorphism. Let Sa=x∈A: x&lt;Aa≠∅ And…

Q: (b) Show that if : R→ S is only assumed to be a ring homomorphism, then it is possible to have a…

A: We can prove the above result using two contradictory examples. Note that these examples are only…

Q: 12. Show that the set of all constant polynomials in Z[r] is a subring but not an ideal of Z[x]

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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…

Q: Let E be the set of all positive even integers. Prove that E is countably infinite by defining a map…

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