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Correspondence theorem

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(a) The kernel of this homomorphism is the principal ideal (x-1). Therefore, Z[x]/(x-1) is isomorphic to Z. According to the correspondence theorem, ideals of Z[x]/(x-1) are in one-to-one correspondence with ideals of Z[x] containing (x-1). Taking into account the above-mentioned isomorphism, we obtain that ideals of Z are in one-to-one correspondence with ideals of Z[x] containing (x-1).
It is well-known that Z[x]/(x2+1) is isomorphic to Z[i]. The correspondence theorem implies that ideals of Z[i] are in one-to-one correspondence with ideals of Z[x] containing (x2+1).

(b) The ring Z[x]/(2x-1) is isomorphic to Z[1/2]. Therefore Z[x]/(6,2x-1) is isomorphic to Z6[1/2].
For Z[x]/(2x2-4,4x-5), we note that (2x2-4,4x-5) =(2, x2-2, 4x-5). Z[x]/(2) is isomorphic to Z2[x]. Under this isomorphism (x2-2) is mapped to x2, while (4x-5) to 1. But Z2[x]/(1) is isomorphic to 0. Therefore Z[x]/(2x2-4,4x-5) is also isomorphic to 0.

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(a) The kernel of this homomorphism is the principal ideal (x-1). Therefore, Z[x]/(x-1) is isomorphic to Z. According to the correspondence theorem, ideals of Z[x]/(x-1) are in one-to-one correspondence with ideals of Z[x] ...

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