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Prove that every infinite finitely generated rng $$ <R,+,.> $$ has an infinite subrng $$ <S,+,.> $$ such that $$ R\not= S $$

Now I know that this claim is false for infinitely generated rngs but I am not sure how to prove this case (I am not actually that certain that it is true)

Solution-rough

As R is finitely generated, denote the generators $$ \{a_1,a_2,.....,a_n\} $$, we can pick a generator with infinite order, wlog take $$ a_1 $$.

Then we can generate the group $$ <0, 2a_1, 4a_1,.....> $$. This is obviously a group under addition however I am not sure how to show that it is closed under multiplication?(or if it even is)

Sorry for the vague explanation;

Thanks for any help

Now I know that this claim is false for infinitely generated rngs but I am not sure how to prove this case (I am not actually that certain that it is true)

Solution-rough

As R is finitely generated, denote the generators $$ \{a_1,a_2,.....,a_n\} $$, we can pick a generator with infinite order, wlog take $$ a_1 $$.

Then we can generate the group $$ <0, 2a_1, 4a_1,.....> $$. This is obviously a group under addition however I am not sure how to show that it is closed under multiplication?(or if it even is)

Sorry for the vague explanation;

Thanks for any help

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