ations ▁of ▁a ▁ring ▁. ▁A ▁connection ▁on ▁an ▁A - module ▁ ▁is ▁defined ▁as ▁an ▁A - module ▁morph ism ▁▁▁▁ ▁such ▁that ▁the ▁first ▁order ▁differential ▁operators ▁ ▁on ▁ ▁obey ▁the ▁Le ib n iz ▁rule ▁▁▁▁ ▁Con ne ctions ▁on ▁a ▁module ▁over ▁a ▁commut ative ▁ring ▁always ▁exist . ▁ ▁The ▁curv ature ▁of ▁the ▁connection ▁ ▁is ▁defined ▁as ▁the ▁zero - order ▁differential ▁operator ▁▁▁▁ ▁on ▁the ▁module ▁ ▁for ▁all ▁. ▁ ▁If ▁ ▁is ▁a ▁vector ▁bundle , ▁there ▁is ▁one - to - one ▁correspond ence ▁between ▁linear ▁connections ▁ ▁on ▁ ▁and ▁the ▁connections ▁ ▁on ▁the ▁- module ▁of ▁sections ▁of ▁. ▁Str ict ly ▁speaking , ▁ ▁corresponds ▁to ▁the ▁cov ari ant ▁differential ▁of ▁a ▁connection ▁on ▁. ▁ ▁Gr aded ▁commut ative ▁algebra ▁▁ ▁The ▁notion ▁of ▁a ▁connection ▁on ▁modules ▁over ▁commut ative ▁rings ▁is ▁straightforward ly ▁extended ▁to ▁modules ▁over ▁a ▁grad ed ▁commut ative ▁algebra . ▁This ▁is ▁the ▁case ▁of ▁super conne ctions ▁in ▁super geometry ▁of ▁grad ed ▁manif olds ▁and ▁super vector ▁bund les . ▁Super conne ctions ▁always ▁exist . ▁ ▁Non comm ut ative ▁algebra ▁▁ ▁If ▁ ▁is ▁a ▁non comm ut ative ▁ring , ▁connections ▁on ▁left ▁and ▁right ▁A - modules ▁are ▁defined ▁similarly ▁to ▁those ▁on ▁modules ▁over ▁commut ative ▁rings . ▁However ▁these ▁connections ▁need ▁not ▁exist . ▁ ▁In ▁contrast ▁with ▁connections ▁on ▁left ▁and ▁right ▁modules , ▁there ▁is ▁a ▁problem ▁how