Theorem. Let M, N, and P be R-modules over a commutative ring 3.2 R. Then (i) MORN = NORM as R-modules. (ii) (MORN)®RP=M®R(N®RP) as R-modules.

Q: Show that for any two finite-dimensional L-representations V and W of L: (a) Hom (V, k) ~V* as…

A: To show that for any tow finite-dimensional L-representation V and W of L HomkV, W≃V*⊗W as L-module

Q: Modern Algebra Let S = {[_$$]:q₂bER} -b Show that : CS given by a ,bER CM₂ (R) a (a + bi) = is a…

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Q: (15 points) Prove the ring Z[√−2] = {a+b√−2 | a,b ≤ Z} is a Euclidean domain with norm 8(a+b√2) = a²…

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Q: Let us define two new operations on R called ~ and ∗ by the following formulas. For real numbers a…

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Q: (4)Let K be integer ring module 12 and let I=([4]) and J=([6]) be ideals of K. Then [2] belong to…

A: Given that, K=Z12 and ideals I and J are such that I=([4])=4Z12={0,4,8} J=([6])=6Z12={0,6} Now,…

Q: {6 ):a € Z,6b,c e Q}. Together with matrix addition and multiplica- Let R= tion, show that Ris a…

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Q: (2) (C) Show that C = 6 ||abER} a, commutative ring with unity even if M₂ (R) is not. Finally, show…

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Q: 3. Let R be a commutative ring. Let x, y E R such that their characteristics are p. Show that (x +…

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Q: Q5:(A) Let R=(Z12 ,+12, -12 ). Find a) Maximal ideals of R b) Is (Z12, +12) a simple group? c) Is…

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Q: (d) Show that the division algorithm holds in Z[V2]. Hint: Let a, 3 e Z[v2] with B+ 0. Write a/3 =r+…

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Q: Find all idempotent elements of the ring M_2(R).

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Q: Show that for any two finite-dimensional L-representations V and W of L: (a) Hom (V, k) ~ V* as…

A: As per bartleby guidelines for more than one question asked only one is to be answered please upload…

Q: Show that submodules of a module of finite length is of finite length. What can you say about the…

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Q: Q2 Let (R, *0)be aring where (asb)* *(czd)=(a+c,b+d) and 2. (asb)o(e)d)= (acpbd, ad td) V…

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Q: Let R be a ring. Show that an R-module M is isomorphic to direct sum of A1 and A2 if and only if…

A: Introduction: A ring is a set equipped with two binary operations, usually called addition and…

Q: 1. The product of 3x²_5x² + 3 and 2x + 5x – 4 in z,[x]/ is а) 2x b) 2x + 1 c) 2x + 2 d) 2x + 3

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Q: Question - and V let Let "M" be an R-Module A, B are sub-Modules of M², then prove that the Quotient…

A: Given : M be an R-Module and A and B are submodules of M. To prove :  AA∩B ≅ A+BB

Q: . Consider the ring Z with ideal . Give the ring tables of Z/ .

A: Consider the ring  with ideal HereLet R = We know thatAndWe have to write the addition and…

Q: Considet the quadratic integer ring Z[sqrt{D}] where D is a square free integer (a) Show 3 is…

A: Given , ℤD  is a ring . (a).  3 is reducible in  D=-2 yet irreducible in  D=-5 . (b).  Is 3 is prime…

Q: Show that a non-abelain Lie algebra of dimension two has precisely three ideals.

A: Show that a non-abelain Lie algebra of dimension two has precisely three ideals.

Q: 5) Suppose that (R, +,.) be a ring without identity and has a subring with identity, then (a) R…

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Q: ! (9) Let K be integer ring module 12 and let I-([4]) and J-([6]) be ideals of K. Then ([0]) (i) I +…

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Q: B E None (i), (ii) (iii), (iv) Which of the following are ring homomorphisms? (i): f: (Q(√2), +,-) →…

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Q: Let K be integer ring module 12 and let I=([4]) and J-([6]) be ideals of K. Then ([0])

A: Let * be integer ring module 12.  Let  I=4J=6

Q: Every submodule of a noetherian (artinian) module is noetherian (artinian).

A: Every submodule of a noetherian (artinian) module is noetherian (artinian).

Q: Show that the power set of 2, 2º, is algebra on 2. an

A: A σ-algebra is a type of algebra so, every sigma algebra is an algebra.        (1)

Q: Compute the Killing form for the Heisenberg algebra and find its radical.

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Q: By the Chinese Remainder Theorem, there is a ring isomorphism y: Z/10Z ~ Z/2Z × Z/5Z, (a) Write out…

A: Answer Explanation:Step 1: Step 2: Step 3: Step 4:

Q: How to show that: If M is a finitely generated R-module and mM = M, then M = 0.

A: How to show that: If M is a finitely generated R-module and mM = M, then M = 0.

Q: 7. Aring with identity (R. +..) is commutative if: (a) Ris an integral domain () (a+ b) - a + Zab +…

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Q: (a) Let S {C ): a, 6, c e Z, where 0 denotes the usual integer zero. Given that S is a ring under…

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