▁a ▁ring ▁R ▁is ▁sem is im ple ▁if ▁and ▁only ▁if ▁all ▁of ▁its ▁modules ▁are ▁sem is im ple , ▁and ▁since ▁sem is im ple ▁modules ▁are ▁preserved ▁under ▁Mor ita ▁equivalence , ▁an ▁equivalent ▁ring ▁S ▁must ▁also ▁have ▁all ▁of ▁its ▁modules ▁sem is im ple , ▁and ▁therefore ▁be ▁a ▁sem is im ple ▁ring ▁itself . ▁ ▁Sometimes ▁it ▁is ▁not ▁immediately ▁obvious ▁why ▁a ▁property ▁should ▁be ▁preserved . ▁ ▁For ▁example , ▁using ▁one ▁standard ▁definition ▁of ▁von ▁Ne umann ▁regular ▁ring ▁( for ▁all ▁a ▁in ▁R , ▁there ▁exists ▁x ▁in ▁R ▁such ▁that ▁a   =   ax a ) ▁it ▁is ▁not ▁clear ▁that ▁an ▁equivalent ▁ring ▁should ▁also ▁be ▁von ▁Ne umann ▁regular . ▁ ▁However ▁another ▁form ulation ▁is : ▁ ▁a ▁ring ▁is ▁von ▁Ne umann ▁regular ▁if ▁and ▁only ▁if ▁all ▁of ▁its ▁modules ▁are ▁flat . ▁ ▁Since ▁flat ness ▁is ▁preserved ▁across ▁Mor ita ▁equivalence , ▁it ▁is ▁now ▁clear ▁that ▁von ▁Ne umann ▁regular ity ▁is ▁Mor ita ▁invariant . ▁ ▁The ▁following ▁properties ▁are ▁Mor ita ▁invariant : ▁simple , ▁sem is im ple ▁von ▁Ne umann ▁regular ▁right ▁( or ▁left ) ▁No ether ian , ▁right ▁( or ▁left ) ▁Art in ian ▁right ▁( or ▁left ) ▁self - inject ive ▁quasi - F ro ben ius ▁prime , ▁right ▁( or ▁left ) ▁primitive , ▁semi prime , ▁semi prim itive ▁right ▁( or ▁left ) ▁( sem