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 mapping, then there exists a unique R-module homo- morphism : M→N such that 18 = f.

Q: Let R be a ring in which each element is idempotent. Let R = R XZ₂. Define + and x on R by (a, [n])…

A: Given: R is a ring in which each element is idempotent and R=R×Z2.To show: + and × are well-defined…

Q: Let R be a regular ring with 1. (i) Prove that for any a E R, there exists an idempotent e ER such…

A: As per the question, we consider regular rings with unity. We prove two results related to…

Q: If Pn is the space of polynomials of degree at most n with a basis B = {1, x, x², ..., xª}, compute…

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Q: A) Find MS (B,C) B) Find a basis for the kernel of S C) Find a basis for the range of S D)…

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Q: Find every possible ring homomorphism from Z15 to Z25 . For each one, compute the kernel and image.

A: To find all the possible ring homomorphisms from Z15 to Z25, we need to consider where the generator…

Q: 3. Find a basis for the vector space RX] of all polynomials in X having degree at most 4 such that…

A: We have to find a basis for the vector space R4[x] of all polynomials in X having degree atmost 4…

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

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Q: -se A, B are submodules (that is, they are isom : M/A→M/B and show pism, then use the FHT

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Q: Let T be ring containing elements e, f, both # 0T, such that e + f = 1r , e² = e, f² = f , and e · f…

A: We are given e+f = 1T,e2=e,f2=f and e.f=0T where T is the rings containing elements e,f both not…

Q: Let R be the ring Z[V3] = {a+b/3|a, b E Z}. For each of the following elements of R determine if it…

A: let R be the ring ℤ3=a+b3|a,b∈ℤ claim- for each element of R determine if it is an invertible…

Q: Let R be a field of real numbers and M₂(R) be a ring of 2 x 2 matrices over R under usual…

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Q: {[: :]we2} b a, b E Z} and let o : R → Z be defined by ø a a Let R = a a а — b. (1) Show that ø is a…

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Q: Consider the set of matrices S = -{(a+b) a + b) : ab € R } Consider the function f: S→ R defined as…

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Q: 4[X]- = 9x₁x₂. representation P for L relative to B₁ and B₂ such that [L(u)]₂=P[u]₁ Let L: R2 R2 be…

A: First of all we find the image of basis elements of domain and write them as a linear combination of…

Q: Suppose X and Y are rings and that f : X → Y is an isomorphism. Let X2x2 be the set of all 2 × 2…

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Q: Let R = R[x] be the ring of real polynomials in the variable x, and let I = be the ideal of R…

A: Given is the ring of real polynomial in the variable of  is the ideal of  generated by For integer ,…

Q: Prove That Let X and Y be two fuzzy rings of R and R' respectively and f: R R is a homo. Then 1- if…

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Q: Let R be a commutative ring and M₂(R) the ring of 2-by-2 matrices with entries in R (with the usual…

A: In the given proof, we showed that if I is an ideal in a commutative ring R, then the set M2​(I) of…

Q: Let M be an R-module, and let N be an R-submodule  of M. Then M is noetherian (artinian) if and only…

A: Given:M is an R-module, and N is an R-submodule of M. To Prove:M is Noetherian (Artinian) if and…

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)

Q: Let f: Z6→ Z2 × Z3 be defined by the following specification of its values 0 → (0, 0) , 1 → (1, 1) ,…

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Q: Let V = span {1+ 2x – x² + 3r³, -2+x+x² – x*, -1+ 8r – x² + 7r", 3 + x – 2r² + r'}. Find a basis ß…

A: Step:-1 Note:- To span V we need the polynomials of degree 3, so dimension of V is 4. (think using…

Q: Let R be a ring and M a left R-module. Show that, for any idempotent e ER End(M) We have ker(e) =…

A: Step 1: Given thatker(e)⊆im(Id−e)im⁡(Id−e)⊆ker(e) Step 2:ker(e)⊆im(Id−e)Take x in ker(e). This means…