user-1: UPDATE
I have managed to get rid of the WARNING: Communication failure, is the MFRC522 properly connected?. Unfortunately, that did not fix the firmware error it now says 0x80. I have now also tried using different IDEs from arduino in the hopes of fixing this, still nothing. The connection error between the reader and the arduino has come back as well.
Furthermore...
*****************************
MFRC522 Digital self test
*****************************
Firmware Version: 0x92 = v2.0
-----------------------------
Only known versions supported
-----------------------------
Performing test...
-----------------------------
Result: DEFECT or UNKNOWN

would this now mean that my device is a defect?
Also, that version has gone back to showing unknown and the other hexadecimal codes.
I have just bought the parts. The arduino nano v3 and the MFRC522 based RFID reader. I have followed the typical pin layout after downloading the library from arduino. When I go to test one of the examples I get a warning within the arduino serial monitor : 
Firmware Version: 0xFF = (unknown)
WARNING: Communication failure, is the MFRC522 properly connected?
Scan PICC to see UID, SAK, type, and data blocks...

I try to scan one of the existing cards but nothing happens. It either stats that it is version 0xFF or 0x0. I am new with this and this is my first 'project'. the pins are defined.

I have 3.3V on the reader to 3v3 on the arduino
I have RST on the reader to D9 on the arduion
I have GND on the reader to GND on the arduino (on the same side as the 3v3 is )
I have MISO on the reader to D12 on the arduino
I have MOSI on the reader to D11 on the arduino
I have SCK on the reader to D13 (on the same side as the 3v3 is)
I have SDA on the reader to D10 on the arduino
The code I am using is from the arduino library manager:
  /*
  * --------------------------------------------------------------------------------------------------------------------
  * Example sketch/program to test your firmware.
  * --------------------------------------------------------------------------------------------------------------------
  * This is a MFRC522 library example; for further details and other examples see: https://github.com/miguelbalboa/rfid
  * 
  * This example test the firmware of your MFRC522 reader module, only known version can be checked. If the test passed
  * it do not mean that your module is faultless! Some modules have bad or broken antennas or the PICC is broken.
  * NOTE: for more informations read the README.rst
  * 
  * @author Rotzbua
  * @license Released into the public domain.
  * 
  * Typical pin layout used:
  * -----------------------------------------------------------------------------------------
  *             MFRC522      Arduino       Arduino   Arduino    Arduino          Arduino
  *             Reader/PCD   Uno/101       Mega      Nano v3    Leonardo/Micro   Pro Micro
  * Signal      Pin          Pin           Pin       Pin        Pin              Pin
  * -----------------------------------------------------------------------------------------
  * RST/Reset   RST          9             5         D9         RESET/ICSP-5     RST
  * SPI SS      SDA(SS)      10            53        D10        10               10
  * SPI MOSI    MOSI         11 / ICSP-4   51        D11        ICSP-4           16
  * SPI MISO    MISO         12 / ICSP-1   50        D12        ICSP-1           14
  * SPI SCK     SCK          13 / ICSP-3   52        D13        ICSP-3              15
  */

 #include <SPI.h>
 #include <MFRC522.h>

 constexpr uint8_t RST_PIN = 9;     // Configurable, see typical pin layout    above
 constexpr uint8_t SS_PIN = 10;     // Configurable, see typical pin layout    above

 MFRC522 mfrc522(SS_PIN, RST_PIN);  // Create MFRC522 instance

 /**
  * Check firmware only once at startup
  */
 void setup() {
   Serial.begin(9600);   // Initialize serial communications with the PC
   while (!Serial);      // Do nothing if no serial port is opened (added for    Arduinos based on ATMEGA32U4)
   SPI.begin();          // Init SPI bus
   mfrc522.PCD_Init();   // Init MFRC522 module

Serial.println(F("*****************************"));
Serial.println(F("MFRC522 Digital self test"));
Serial.println(F("*****************************"));
mfrc522.PCD_DumpVersionToSerial();  // Show version of PCD - MFRC522 Card Reader
Serial.println(F("-----------------------------"));
Serial.println(F("Only known versions supported"));
Serial.println(F("-----------------------------"));
Serial.println(F("Performing test..."));
bool result = mfrc522.PCD_PerformSelfTest(); // perform the test
Serial.println(F("-----------------------------"));
Serial.print(F("Result: "));
if (result)
  Serial.println(F("OK"));
else
  Serial.println(F("DEFECT or UNKNOWN"));
Serial.println();
 }

 void loop() {} // nothing to do


user39658: You are using SPI comm. The code is fine too. Usually this "communication failure" happens due to wrong pin connections. Say for eg. If MOSI and MISO are interchanged. Check all connections, ground and VCC connections once more. I have also seen that soldering the pins of MFRC522,  instead of simply connecting by wire has solved this issue most of the time. Another work around is repeatedly calling PCD_init() function even after failing result inside loop() . And see whether its working. If none of these worked, hard luck. I think its firmware issue or defective board.

-1: <p><strong>UPDATE</strong></p>

<p>I have managed to get rid of the <code>WARNING: Communication failure, is the MFRC522 properly connected?</code>. Unfortunately, that did not fix the firmware error it now says <code>0x80</code>. I have now also tried using different IDEs from arduino in the hopes of fixing this, still nothing. The connection error between the reader and the arduino has come back as well.</p>

<p>Furthermore...</p>

<pre><code>*****************************
MFRC522 Digital self test
*****************************
Firmware Version: 0x92 = v2.0
-----------------------------
Only known versions supported
-----------------------------
Performing test...
-----------------------------
Result: DEFECT or UNKNOWN
</code></pre>

<p>would this now mean that my device is a defect?</p>

<p>Also, that version has gone back to showing <code>unknown</code> and the other hexadecimal codes.</p>

<p>I have just bought the parts. The arduino nano v3 and the MFRC522 based RFID reader. I have followed the typical pin layout after downloading the library from arduino. When I go to test one of the examples I get a warning within the arduino serial monitor : </p>

<pre><code>Firmware Version: 0xFF = (unknown)
WARNING: Communication failure, is the MFRC522 properly connected?
Scan PICC to see UID, SAK, type, and data blocks...
</code></pre>

<p>I try to scan one of the existing cards but nothing happens. It either stats that it is version <code>0xFF</code> or <code>0x0</code>. I am new with this and this is my first 'project'. the pins are defined.<a href="https://i.stack.imgur.com/xOVCq.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/xOVCq.png" alt="enter image description here"></a></p>

<p><a href="https://i.stack.imgur.com/JW89P.jpg" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/JW89P.jpg" alt="enter image description here"></a></p>

<p>I have 3.3V on the reader to 3v3 on the arduino
I have RST on the reader to D9 on the arduion
I have GND on the reader to GND on the arduino (on the same side as the 3v3 is )
I have MISO on the reader to D12 on the arduino
I have MOSI on the reader to D11 on the arduino
I have SCK on the reader to D13 (on the same side as the 3v3 is)
I have SDA on the reader to D10 on the arduino</p>

<p>The code I am using is from the arduino library manager:</p>

<pre><code>  /*
  * --------------------------------------------------------------------------------------------------------------------
  * Example sketch/program to test your firmware.
  * --------------------------------------------------------------------------------------------------------------------
  * This is a MFRC522 library example; for further details and other examples see: https://github.com/miguelbalboa/rfid
  * 
  * This example test the firmware of your MFRC522 reader module, only known version can be checked. If the test passed
  * it do not mean that your module is faultless! Some modules have bad or broken antennas or the PICC is broken.
  * NOTE: for more informations read the README.rst
  * 
  * @author Rotzbua
  * @license Released into the public domain.
  * 
  * Typical pin layout used:
  * -----------------------------------------------------------------------------------------
  *             MFRC522      Arduino       Arduino   Arduino    Arduino          Arduino
  *             Reader/PCD   Uno/101       Mega      Nano v3    Leonardo/Micro   Pro Micro
  * Signal      Pin          Pin           Pin       Pin        Pin              Pin
  * -----------------------------------------------------------------------------------------
  * RST/Reset   RST          9             5         D9         RESET/ICSP-5     RST
  * SPI SS      SDA(SS)      10            53        D10        10               10
  * SPI MOSI    MOSI         11 / ICSP-4   51        D11        ICSP-4           16
  * SPI MISO    MISO         12 / ICSP-1   50        D12        ICSP-1           14
  * SPI SCK     SCK          13 / ICSP-3   52        D13        ICSP-3              15
  */

 #include &lt;SPI.h&gt;
 #include &lt;MFRC522.h&gt;

 constexpr uint8_t RST_PIN = 9;     // Configurable, see typical pin layout    above
 constexpr uint8_t SS_PIN = 10;     // Configurable, see typical pin layout    above

 MFRC522 mfrc522(SS_PIN, RST_PIN);  // Create MFRC522 instance

 /**
  * Check firmware only once at startup
  */
 void setup() {
   Serial.begin(9600);   // Initialize serial communications with the PC
   while (!Serial);      // Do nothing if no serial port is opened (added for    Arduinos based on ATMEGA32U4)
   SPI.begin();          // Init SPI bus
   mfrc522.PCD_Init();   // Init MFRC522 module

Serial.println(F("*****************************"));
Serial.println(F("MFRC522 Digital self test"));
Serial.println(F("*****************************"));
mfrc522.PCD_DumpVersionToSerial();  // Show version of PCD - MFRC522 Card Reader
Serial.println(F("-----------------------------"));
Serial.println(F("Only known versions supported"));
Serial.println(F("-----------------------------"));
Serial.println(F("Performing test..."));
bool result = mfrc522.PCD_PerformSelfTest(); // perform the test
Serial.println(F("-----------------------------"));
Serial.print(F("Result: "));
if (result)
  Serial.println(F("OK"));
else
  Serial.println(F("DEFECT or UNKNOWN"));
Serial.println();
 }

 void loop() {} // nothing to do
</code></pre>
 39658: <p>You are using SPI comm. The code is fine too. Usually this "communication failure" happens due to wrong pin connections. Say for eg. If MOSI and MISO are interchanged. Check all connections, ground and VCC connections once more. I have also seen that soldering the pins of MFRC522,  instead of simply connecting by wire has solved this issue most of the time. Another work around is repeatedly calling PCD_init() function even after failing result inside loop() . And see whether its working. If none of these worked, hard luck. I think its firmware issue or defective board.</p>
