The ideal generated by a subset $T$ of a ring $R$ is the smallest ideal containing $T$.
Equivalently, $(T) = \bigcap\limits_{I\supseteq T} I$ where $I$ varies over all of the ideals in $R$ that contain $T$.
Equivalently, if $T = {t_i}{i=1}^n$, then $(T) = \left{\sum\limits{i=1}^n r_it_i\right}$ , the set of all formal sums of elements of $T$ with coefficients in $R$.