Def/Ideal generation



=Construction= Suppose that $R$ is a commutative ring, and $S$ is a subset of $R$. Let $M$ be the set of ideals of $R$ which contain $S$ as a subset.

Then, $M$ is nonempty, since $R$ is an ideal of $R$ which contains $S$, and hence $R \in M$.

Define $ \langle S \rangle = \bigcap M$ (the intersection of all ideals containing $S$). Then, we find that:
 * $\langle S \rangle$ is an ideal in $R$, since intersections of ideals are ideals.
 * If $I$ is an ideal of $R$, and $I$ contains $S$ (so that $I \in M$), then $I$ contains $\langle S \rangle$, using basic properties of intersection.

=Construction= Suppose that $R$ is a commutative ring, and $S$ is a finite subset of $R$. If $n$ is a natural number, for which $S$ has $n$ elements, we may write: $$S = \{ s_1, \ldots, s_n \}.$$ In this case, the ideal $\langle S \rangle$ can be described as: $$\langle S \rangle = \{ r \in R \mbox{ such that } \exists r_1, \ldots, r_n, s = \sum_{i=1}^n r_i s_i \}.$$ In other words, $\langle S \rangle$ is the set of $R$-linear combinations of elements of $S$.

Indeed, $\langle S \rangle$, as described above, is certainly an ideal containing $S$. Furthermore, every ideal containing $S$ contains all expressions of the form $\sum_{i=1}^n r_i s_i$. Hence $\langle S \rangle$, as described above, is minimal among ideals containing $S$, as desired.

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