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Ring Homomorphisms and Ideals

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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. Non-trivial ideas are proved or disproved.

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1. 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.

It is not an ideal. ...

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