The battery problem indicator light is on. I have

Without having the vehicle an doing some testing we'd only be guessing . Your best bet would be have a qualified repair shop look at it . You replaced the starter but the battery light stays on ? That's because the starter has nothing to do with the charging system . The battery light is an indication the charging system has a problem . The charging system on your vehicle is controlled by the PCM - engine computer . The first thing to do would have been to hook up a scan tool an check for codes in the PCM & BCM .

brake indicator could be low reservoir level
take it back to where you got the alternator and have it all tested
it could be a problem voltage regulator that is causing a problem in the ECM resulting in the lights

Regulator in the alternator. If you can get one, replace it. If not replace the alternator. I believe yours is to old to have the regulator in the ECM. You might want to check this.

Poor connection either at the starter, battery, alternator, or loose engine ground strap. Fully charge the battery also.

starter solenoids only work when the ignition switch is in the start position. Otherwise there is no power to it so the solenoid would not figure in this fault.. The indications are loose battery wires or a dead battery. have the battery load tested and check the wiring for the alternator is in good condition and tight.

yes it could.If the alternator is bad your battery light should come on while running.when u replaced the battery it should have ran good for a minute till the alternator started draining draining your battery.Then the battery light would come on.I would replace it or have it checked.You can check it by starting it up and keep any elelectric off radio,lights,ect.then disconnect the battery if the car stalls its the alternator.the car would stay running with good alternator

Sounds like the starter battery can't hold the charge from the alternator. A 1 hour highway drive should charge the battery enough - if it's the same the next morning it's up for replacement. Assuming there are no lights, radio or other devices on in the car during the night.

If the battery has been recently replaced - the alternator might not supply enough current anymore. In this case alternator brushes might need replacement or the voltage regulator has died. Usually the battery indicator lights up in this case. The battery indicator light can be checked by turning the ignition key one notch so all lights in the dash light up. If the battery indicator doesn't light the lightbulb might be dead.

Battery light indicates alternator failure. Alternator may have internal short or perhaps something else in the car is shorted. Can you jump start?

run scanner for trouble codes, some auto parts store will do this for free, check starter solenoid, check voltage at battery , 12+volts not started ,  14+ amps when running 

If your alt is putting out 12V you have a faulty alternator. A healthy alternator puts out 14v not 12v. The battery light doesn't indicate a bad battery, rather it indicates voltage is below 14v or 13v, depending on the vehicle.

With the vehicle running use a voltmeter at the alt and test for output voltage, if it's any where close to 12v you have a faulty alt and will need to be replaced.

You could also have a corroded/faulty/loose battery cable/s going from the alt to the starter/battery/alt.