Editing Radical of a ring (section)

=== Singular radical ===
An element of a (possibly [[non-commutative ring|non-commutative]] ring) is called left '''singular''' if it annihilates an [[essential submodule|essential]] left ideal, that is, ''r'' is left singular if ''Ir'' = 0 for some essential left ideal ''I''.  The set of left singular elements of a ring ''R'' is a two-sided ideal, called the [[singular submodule|left singular ideal]], and is denoted <math>\mathcal{Z}(_R R)</math>.  The ideal ''N'' of ''R'' such that <math>N/\mathcal{Z}(_R R)=\mathcal{Z}(_{R/\mathcal{Z}(_R R)} R/\mathcal{Z}(_R R))\,</math> is denoted by <math>\mathcal{Z}_2(_R R)</math> and is called the '''singular radical''' or the '''Goldie torsion''' of ''R''.  The singular radical contains the prime radical (the nilradical in the case of commutative rings) but may properly contain it, even in the commutative case.  However, the singular radical of a [[Noetherian ring]] is always nilpotent.