Operations on generators of an ideal

Suppose I have an ideal $I = (r, s, t)$. What operations can I apply to the $r$, $s$, and $t$, so that I get an ideal equal to $I$? For example, $(r, s, t) = (r - s, s, t)$. What are the other ones?

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HINT $\$ In analogy with change of basis in linear algebra, consider linear transformations of the ideal generators $\rm\: (r',s',t') = (r,s,t)\:A\:.\:$ What kind of matrices $\rm\:A\:$ suffice to preserve the ideal?