INSTRUCTOR PLT OPERATED A C152 WHEN THE ACFT REGISTRATION CERTIFICATE DID NOT SHOW THE ACTUAL LEGAL OWNER.

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I WAS PHOTOGRAPHED BY AN FAA INSPECTOR AS I TAXIED BACK TO OUR FLT SCHOOL. UPON ARR AT THE RAMP HE ASKED FOR MY FLT CERTIFICATES AND MEDICAL; AFTER WHICH HE INFORMED ME THAT THE REGISTRATION OF THE ACT HAD AN ANOMALY AND WAS THEREFORE ILLEGAL TO FLY. I EXPLAINED TO HIM (FAA INSPECTOR) THAT I HAD THE ACFT'S REGISTRATION ON BOARD THE ACFT; BUT HE EXPLAINED THAT THE REGISTERED OWNER/COMPANY HAD GONE OUT OF BUSINESS AND WAS THEREFORE NON EXISTENT. I THEN EXPLAINED THAT I HAD NO IDEA THAT THE COMPANY WAS OUT OF BUSINESS; TO WHICH HE RESPONDED BY TELLING ME THAT THE OPERATOR OF THE ACFT HAD BEEN INFORMED TWICE ALREADY. BY THIS TIME THE SCHOOL'S OPERATOR HAD JOINED US AND ADMITTED TO NOT TELLING/INFORMING ME ABOUT THE ACFT'S REGISTRATION PROB. THE FAA INSPECTOR INFORMED ME THAT AS PIC I WOULD BE VIOLATED SHOULD THE INVESTIGATION PROVE THAT THE ACFT IS IMPROPERLY REGISTERED. HE LATER SAID A LETTER OF WARNING WOULD MOST LIKELY BE ISSUED IN THIS CASE; SINCE I WAS IGNORANT OF THE SIT. HE ADVISED THAT IN THE FUTURE I SHOULD MAKE EXTENSIVE INQUIRIES ABOUT AN ACFT'S DOCUMENTATION BEFORE FLYING IT. I ADMIT TO NOT MAKING EXTENSIVE INQUIRIES AS THE OPERATOR INFORMED ME THAT THE ACFT WAS ON A LEASE BACK PURCHASE AND THE REGISTRATION WOULD BE DONE IN HIS NAME AS SOON AS THE LAST PAYMENT WAS MADE. HE ALSO SAID THAT HE THOUGHT FROM HIS DISCUSSION WITH THE FAA INSPECTOR THE ISSUE WAS NOT SERIOUS ENOUGH TO WARRANT GNDING OF THE ACFT. THE ACFT HAS SINCE BEEN GNDED UNTIL THE MATTER IS RESOLVED.