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Ideal (Ring Theory)**

Given a ring \((R,+,\times)\), a subset \(I\subseteq R\) such that \((I,+)\) is a subgroup of \((R,+)\) and \(RI=\{ra|r\in R,a\in I\}\subseteq I\) (resp. \(IR=\{ar|r\in R,a\in I\}\subseteq I\)) is called a left (resp. right) ideal of \(R\). If \(I\) is both a left and right ideal then we call it a two-sided ideal and write \(I\triangleleft R\). An ideal is called 'proper' if it is not \(R\) itself, and 'nontrivial' if it is not the zero ideal \(0=\{0_R\}\).