Why is Satan commonly believed to rule in hell? Satan (Devil, Lucifer and all these not exactly the same meaning names) is often told to rule in hell, tormenting the damned souls and so on, while in the Bible I could not find anything more than the fact of putting him there. Basing only on the Bible I'd rather assume that he is as punished there together with human sinners.

Where does the idea of Satan as the ruler of hell and punisher of the souls come from and is it an official doctrine (in either Catholic or Protestant Churches)?
I can only answer from a Protestant perspective and that perspective would be from scripture itself, since Protestants professedly believe that scripture is 'binding' (I am quoting here from the the first of the 39 articles of the C of E for want of anything more broadly indicative.)

I cannot find anything that 'explains' why there is a misconception. But misconception, I agree, there definitely is, through (I would say) ignorance of the scripture itself.

So I would suggest that ignorance is the reason for the misconception rather than any party or parties deliberately putting about an error.

I would suggest that it is just superstitious rumour from non-religious sources and from paintings which attempt to depict hellfire, such as those paintings accompanying the written work of Dante Alighieri.

A previous question dealt with the common misconception that the Entity variously referred to as Serpent, Satan and Lucifer (in Hebrew descriptions); as Diabolos, Poneros and Antidikos (in Greek descriptions); and as 'the Devil' (a rather poor English translation) will have the power to torment the souls and bodies of resurrected humans in the lake of fire.

The first answer to that previous question explained from scripture that the misconception was false. However the second answer to the question introduced the concept of the Entity "ruling" in hell - something that the question had not asked.

So this present question focuses on the matter of 'ruling'.

I would agree with the OP here as I, also, cannot find any text of scripture that attributes future 'rule' (in hell) to the Entity so described above. (They are not personal names for no personal name is ever given to that individual, only descriptive titles that convey his activities and characteristics).

The texts only convey (as the OP states) the fact of the presence of Satan, not any activity open to him to conduct in the future.

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The last - specific - mention in the bible of that individual is in Revelation 20:10 :


  ... the diabolos that deceived them was cast into the lake of fire and brimstone ... [KJV] (I have written 'diabolos' rather than 'devil'.)


Thereafter, in the closing two chapters of scripture, there are two general references in which he would be included with all who are in the same state as he.


Firstly there are those who did not overcome within the context of time and who shall therefore not inherit anything in the new heaven and the new earth, Revelation 21:8. The multiple description ends with the category 'all liars'.



  ... and all liars, shall have their part in the lake of fire which burneth with fire and brimstone: which is the second death. [KJV].



Secondly, there are those have no right to the tree of life and thus may not enter the gates of the new city, Revelation 22:15. This multiple description ends with the category 'and whosoever loveth and maketh a lie'. 


They are described as being :


  without.


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Thus after being described for the last time, the individual under discussion is no longer given any specific attention and is classed among all those who behaved in similar fashion.

Which, I would say, is conclusive proof that the idea of him 'ruling' anything is a fiction.

To those men, who behaved towards Jesus in a way which was exactly the same as the way in which the Entity behaved towards him, he said :


  Ye are of your father the devil, and the lusts of your father ye will do. He was a murderer from the beginning, and abode not in the truth, because there is no truth in him. When he speaketh a lie, he speaketh of his own: for he is a liar, and the father of it. John 8:44 [KJV].


The last chapters of the Apocalypse indicate that both the father of liars and  all who are liars like him, shall be together, none ruling another, for ever in the lake of fire.