Hello!
When I'm trying to run a program (Ex: DumpInfo) it shows me this:

Connection:
SDA-> PIN 10
SCK-> PIN 13
MOSI-> PIN 11
MISO-> PIN 12
GND-> GND
RST-> PIN 9
3.3V-> 3.3V

Could it be a problem with the chip?


That's a 3.3V device. You must not connect the SPI pins directly to the 5V pins on the UNO! You might have already damaged the MFRC522 chip. You must use a level converter to adjust the SPI voltage levels.
The pins are connected at 3.3V on Arduino Uno, not 5V.
sebastian-s:
The pins are connected at 3.3V on Arduino Uno, not 5V.
No, the power pin is connected to 3.3V but the SPI signals connect directly to the corresponding SPI pins on the Arduino (at least you wiring diagram says so) and these have a 5V level. That's why you need a level converter.