Subring of finitely generated rng

[hmmmmm]

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

 

[jvicens]

I appreciate your curiosity and desire to prove this claim. However, as you have mentioned, this claim is actually false for infinitely generated rngs. In fact, it is possible for an infinite finitely generated rng $$ <R,+,.> $$ to not have any infinite subrngs at all.

To prove this, let us consider the rng $$ R = \mathbb{Z}[x] $$, the set of all polynomials with integer coefficients. This rng is finitely generated by the elements $$ \{1,x\} $$, but it does not have any infinite subrngs. This is because any subrng of $$ R $$ must contain only polynomials with integer coefficients, and therefore can only have a finite number of elements.

In your solution, you have attempted to generate an infinite subrng by taking the generator with infinite order and generating a group using it. However, this group does not necessarily form a subrng of $$ R $$, as it may not be closed under multiplication. For example, if we take the generator $$ a_1 = x $$, then the group you have generated is $$ <0, 2x, 4x,...> $$, which is not closed under multiplication as $$ (2x)(4x) = 8x^2 $$ is not a polynomial with integer coefficients.

In conclusion, while it is true that every infinite finitely generated rng has a subrng, it is not necessarily true that this subrng will be infinite. It is important to carefully consider the definitions and properties of rngs in order to prove claims such as these. I hope this helps clarify the situation for you.