Abstract Algebra Answers

Prove that an ideal M neq R in a commutative ring R with identity is maximal if and only if for every r in R-M, there exists x in R such that 1_R - rx in M.

the set of all functions of {1 2 3...,10} to itself forms a group w.r.t the composition of function.true or false

Let S be the set of all polynomial with real coefficient ,if f, g€S, define f~g if f¹=g¹ where f¹ is the derivative of f, show that ~ is an equivalence relation. Describe the equivalence classes of S

let i=<x,2> and=<x,3> be ideal in z[x] .prove that IJ=<x,6>

If an ideal is contained in union of two ideals then show that it is wholly contained in one of them

Let G be multiplication group all positive real number and R the additive group of all real number. Is G a subgroup of R? explain

Prove or disprove: Every subgroup of the integers has finite index.

Let f : G → G0 be a group epimorphism, and let H be the normal subgroup that be the Kernal of the epimorphism. Then, prove that G0 is isomorphic to G/H.

PROVE that N (a) is a subgroup of G Where N (a) is a normalizer of a in G

F ind aba -1 w here (i) a = (5 , 7 , 9) , b = (1 , 2, 3) (ii) a = (1 , 2,5)(3 , 4) , b = (1 , 4 , 5)