Can someone show the proof of the ideal test?

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I have just began studying ideals and came across the ideal test. The text says that we can apply a subgroup test and the subring test but I uncertain how to start and proceed. Can someone show me this proof?

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1 Answer

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Let $R$ be a ring and $S$ a subset. Then $S$ is by definition an ideal if

1) $S$ is a subgroup of the underlying additive group of $R$.

2) If $r \in R$ and $s \in S$, then $rs, sr \in S$.

To show that $S$ is an ideal, you simply show that these two conditions hold. Apply an appropriate subgroup test to show the first one. For instance, if $S$ is non-empty and for all $x,y \in S$, we have $x-y \in S$, then $S$ is a subgroup.

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