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Let a commutative ring R be generated by {a_1, a_2, ..., a_n}

Let a commutative ring R be generated by {a_1, a_2, ..., a_n}
such that [a_1, a_2, ... , a_n] = {(a_1xr_1) + (a_2xr_2) + ... + (a_nxr_n) for
r_1, ..., r_n in set of Reals}.

I need to show this set is an ideal. Do I just need to show that it satisfies the commutative properties of the ideal?

Solution Preview

Are you viewing R as module over the real numbers (with unity)? or over itself, in which case R is (trivially) an R-module; and ...

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