A C172 INSTRUCTOR AND STUDENT; AFTER LNDG AT SBA; EXITED THE RWY ONTO AN INTERSECTING RWY WITHOUT A CLRNC.

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I WAS GIVING INSTRUCTION TO A STUDENT ON IFR PROCS. WE HAD OBTAINED AN IFR CLRNC TO SBA FROM SMO AND EXECUTED THE ILS 7 APCH TO A LNDG; INTENDING TO TAXI BACK; GET A RETURN IFR CLRNC BACK TO SMO; AND DEPART. AFTER TOUCHING DOWN; I TOLD MY STUDENT ESSENTIALLY; 'IF THEY DON'T TELL US WHAT TO DO; JUST MAKE A R TURNOFF AT THE NEXT TXWY.' AT THIS POINT; WE WERE STILL ON FREQ WITH SBA TWR. THE ACFT ROLLED PAST TXWY F AS TWR INSTRUCTED US TO 'CONTACT GND;' EVEN THOUGH WE WERE STILL ON RWY 7 ROLLING OUT; NOR WERE WE TOLD ANY PREFERRED TURNOFF DIRECTION. MY STUDENT IMMEDIATELY SWITCHED TO GND CTL FREQ AND SAID WE WERE 'WITH THEM' (POOR RADIO TECHNIQUE); AS THE ACFT APCHED RWY 15R'S INTXN. AT THIS TIME THE STUDENT BEGAN A R TURN ONTO RWY 15R; STILL FUMBLING WITH THE RADIO; AND I REALIZED HE INTENDED TO TURN OFF ONTO RWY 15R; PROBABLY MISTAKING IT FOR A TXWY. I STOOD ON THE BRAKES AND BEGAN TO TAKE CTL OF THE PLANE; INTENDING TO SWING IT BACK ONTO RWY 7; AS THE GND CTLR NOTIFIED US THAT WE WERE 'ABOUT TO TAXI ONTO AN ACTIVE RWY WITHOUT PERMISSION.' I GOT THE PLANE STOPPED AND RADIOED THE CTLR THAT WE WERE 'STOPPING HERE; AND HOW WOULD YOU LIKE US TO PROCEED.