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variable current rating vs fixed current rating from dc battery, does both of them exist or only one

i came across batteries with mah rating, which says milli amps amount of current can be supplied for an hour, if we pull more power more discharge for less time, some batteries are rated with voltage and output current, does that means its current output is fixed? for some batteries its output current rating is not mentioned but mah rating and voltage is specified, does that mean current can be discharged according to the resistance in circuit ? i have a gsm module need 9v, and 1 amp current to start transmission, it works with 12v 1.0 amp dc adapter, but when i use a 9v dc battery whos current rating is not mentioned on the battery it starts but turns off as soon as i tried to make call, can i use a mah rating 9v battery, so that the gsm module takes current as its need? if not what can i use to take 1 amp current from the dc battery? is there a way? i'm not a electronics engineer i'm a software guy who doesn't know about electronics. i'm working with bus tracking project for my college, but unfortunately i couldn't find help for electronics thing