# Subring of finitely generated rng

#### hmmm16

##### Member
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

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