On essential left ideals of associative rings (Q2770656)

The authors investigate essential left ideals of associative rings (generally without unit). Recall that a left ideal \(L\) of a ring \(R\) is essential if \(L\cap I\neq 0\) for every nonzero two-sided ideal \(I\) of \(R\), and \(L\) is a direct summand of \(R\) if there exists a two-sided ideal \(I\) of \(R\) such that \(R=L\oplus I\).NEWLINENEWLINENEWLINEIt is shown that the following conditions are equivalent for any ring \(R\): (i) \(R\) has no proper essential left ideal; (ii) each left ideal of \(R\) is a direct summand; (iii) \(R\) is a direct sum of simple rings having only trivial left ideals (Theorem 5). Further, the authors prove that \(R\) has no proper essential left ideals and a right unit iff every left ideal of \(R\) has a unit iff \(R\) is a finite direct sum of division rings (Theorem 7). Finally, rings having an essential minimal left ideal are characterized (Theorem 8).