My mother has a heart murmur from rheumatic fever (?). When she was hospitalized, they said she had arrythmias. Is this the same thing?

No. A murmur is noise the doctor hears when blood flows through a heart valve.An arrhythmia , or dysrhythmia , encompasses changes in the heart rhythm, from a regular even rhythm , to an uneven rhythm.Or a very slow or very fast rhythm.
NO. Not the same thing but they may be related. Valvular heart disease can certainly increase the risk that the patients heart may also have rhythm problems. The rhythm problem is related the the electricity network and pathways and when a valve leaks the chamber behind can get bigger and stretch and damage the electrical pathways.