The smallest ideal containing $S$

Question

In a commutative ring $R$, the ideal generated by the set $S=\{a_1,\dots,a_n\}$, where $S$ is a subset of $R$. $$(a_1,\dots,a_n):= \left\{\sum_{k=1}^n r_ka_k: r_k\in R\right\}$$

is the smallest ideal of $R$ containing $S$.

How can I prove it ?

Answer 1

As @mixedmath stated above, showing $(a_1, \dots, a_n)$ is an ideal is a quick exercise and tests your understanding of the various definitions.

To show $(a_1, \dots, a_n)$ is the smallest ideal of $R$ containing $S = \{a_1, \dots, a_n\}$, you want to show that if $I$ is an ideal of $R$ that contains $S$, then $(a_1, \dots, a_n) \subseteq I$; this is also straightforward.