The file MRT.exe is a genuine Microsoft process file, however it is not supposed to be running continuously. MRT.exe is the process of the Microsoft Malicious Software Removal Tool, an original Microsoft tool that checks your system for malicious software. Thus, the process itself is not a malicious process.
However, MRT.exe usually runs once in every month, unless manually started up. On the first Tuesday of every month is Microsoft’s Patch Tuesday, the day when they release security updates for all their software: Windows, Office and others. On this day, a new version of MRT.exe gets downloaded and runs in the background to check your system.
Whenever it finds something, it will be logged in a log file on the computer. You can check this log by running %systemroot%\debug from the Run window and opening the file called mrt.log in Notepad.
Now, regarding the MRT.exe process: if you find it running in the background, and it is not a Patch Tuesday, then something is probably wrong. Search your entire hard drive for MRT.exe and if you find it any folder other than C:Windows\system32, you should delete it.