Let M be an R-module, and let N be an R-submodule  of M. Then M is noetherian (artinian) if and only if   both N and M/N are noetherian (artinian).

Q: Let R be a ring and let M be a free R-module with basis B. If N is any R-module and f: B→ N is any…

A: Let R be a ring and let M be a free R-module with basis B. If N is any R-module and is any mapping,…

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: Prove that [x] ∈ ℤ/n is a zero divisor if and only if there exists an integer y >1 such that y | x…

A: Let x∈ℤn is a zero divisor. There there exists a nonzero element y such that  xy=0 in ℤn. Hence xy…

Q: Let BCR be a subset of R. Prove that there is an ideal I of R such that B n I has at most one…

A: To show that ideals I and J  exist satisfying conditions 1) and 2)

Q: State and prove Euclid’s Lemma.(you may assume without proof that for integers x and y there exist…

A: If x and y are integer not both zero then there exist integer k and l such that kx+ly=gcdx,y

Q: How large should n be to guarantee that the Trapezoidal Rule approximation to 8 n = 4 (-x¹ + 22x³ -…

A: (a)How large should n be to guarantee that the Trapezoidal Rule approximation  to is accurate to…

Q: prove That Lemma: Let A be a fuzzy ideal of Roah and br be fuzzy singleton. Then (a+b)noA-ahobno A

A: Given Equation:(A∘B)H∘A=AH∘BH∘A

Q: _C 16. Suppose R is commutative and let I and J be ideals of R, so R/I and R/J are naturally…

A:

Q: Show that Zmn is ring-isomorphic to Zm ⨁ Zn when m and n are relativelyprime

A: Given: Zmn and Zm⊕Zn. To show:Zmn is ring-homomorphic to Zm⊕Zn when m and n are relatively prime.

Q: Prove that if x is even and y is even, then the product of x and y is divisible by 4.

A: We use the Division Algorithm to prove the given statement.

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: How to prove that: If M is a finitely generated R-module and {m1,....,mn} in M are elements of M…

A: How to prove that: If M is a finitely generated R-module and {m1,....,mn} in M are elements of M…

Q: Manuben

A: The objective of this question is to prove that the set Z[w] = {a+bw | a, b are integers} is a…

Q: The set H= {0,2,3}is a subring of (Z6, +6,.6) integer T F ring module 6.

A:

Q: Prove That Let X andy be two fuzzy rings of Rand R respectively and fiRR' is home. Then if B is a…

A:

Q: Z[] {a + bi| a, b EZ}. Show that Z[] Let denote the set a subring of C.

A: Here to show z [i] is a subring we need to show show z [i] is closed under substraction and…

Q: Let A = (a, a, as) and B = {b₁,b₂, b3} be bases for V, and let P = {[b₁]A, [b₂]A, [b3]A} (are)…

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Q: Construct a nontrivial homomorphism $: Zg → Zı5 >

A:

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: 2. List all the polynomials in Z3[x] that have degree 2.

A: To find all the polynomials in Z3[x] that have degree 2

Q: If R is a ring and S is a subring of R, then S is  an R-module (module over R).

A: R is ring and S is a subring of R  Claim : S is an R -module (module over R)