Theorem 12. Let R is a commutative ring and r e R,fe Hom (M, M'), then rfe. R M, M') defined by (rf) (x) = rf (x) for all x M and with this scalar multiplication the a oup Hom (M, M') is an R-module. R

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: _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: 4. Let R be a ring. (a) Let x, y, z € R. Prove that, if x | y and y | z, then x | z. (b) Prove that…

A: Let R be  a  ring .4) a)  Let  Suppose that   divides y and  y  divides z.We  have  to show that x…

Q: Consider the quotient ring R1 = Z3[a]/(x² + 1) and R2 = Z3[i] = {a + bi / a,b e Z3} with operations…

A: Consider the quotient ring R1=Z3x/x2+1  and  R2=Z3i=a+bi/a,b∈Z3 with operations…

Q: 1. Based on the axioms of a vector space V over a field F prove the following: (a) If x, y z ¤ V and…

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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: Which of the following are ring (i): f: (2Z, +,-) → (Z, +,-), (ii): ƒ: (Z8, +,-) → (Z4, +,-), - 2x -…

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Q: Let R be a commutative unitary ring and let M be an R-module. For a fixed reR let rM = {rx : x€M}…

A: Given: is a commutative unitary ring and let  be an -module. For a fixed  let and To Prove:a) Show…