I=(x^2-3, x+3) is Ideal of Z[x]

Prove That Z[x]/I  is congruent to Z6


pf ] We need to show that (x, 6) = I

1st  6 = (x^2-3)-(x+3)(x-3)

      x = (x+3)-3

      so (x, 6) ⊂ I is hold


2nd x^2-3 = (x)x-3

   x+3 = (x)+3

   so I ⊂ (x, 6) is hold

So our Proposition is Hold.


By 3rd Isomorphic theorem for Ring theory,

Z[x]/I congruent Z[x]/(x, 6) 

        congruent Z6[x]/(x) 

        congruent Z6 ■


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