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Ring Homomorphisms and Ideals

Let : R->Q be a ring homomorphism , and suppose that I is a non-trivial ideal of R.
Prove or disprove that (I)={ (i)| i I } is necessary an ideal of Q.

Let : R->Q be a ONTO ring homomorphism , and suppose that I is a non-trivial ideal of R.
Prove or disprove that (I)={ (i)| i I } is necessary an ideal of Q.

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Ring Homomorphisms and Ideals are investigated.

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