Definition: Generating Set of an Ideal

Let $(R,\cdot,+)$ be a ring. For every subset $A\subseteq R$ define $$\langle A\rangle_R:=\bigcap_{A\subset I,I\lhd R}I$$ as the set intersection of all ideals $I$ of $R$ containing $A.$

Obviously, $A\subset \langle A\rangle_R$ and $\langle A\rangle_R \subset I$ for every ideal $I$ of $R.$ Therefore, $\langle A\rangle_R$ is the smallest ideal of $R$ containing $A.$ We call $A$ the generating set of $\langle A\rangle_R$ and $\langle A\rangle_R$ the ideal generated by $A.$ If $A$ contains only a finite number of elements, we can also write $\langle a_1,\ldots, a_n\rangle_R.$

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  1. Modler, Florian; Kreh, Martin: "Tutorium Algebra", Springer Spektrum, 2013