Inter Milan will play two home league games behind closed doors following the alleged racist abuse of Napoli defender Kalidou Koulibaly on Wednesday.
Inter Milan will also play a third match at the San Siro without opening the 'curva' section of the ground, popular with fans known as 'ultras'.
Napoli boss Carlo Ancelotti said he asked for the match to be suspended three times and that Koulibaly was targeted in the 1-0 defeat.
Inter are deciding whether to appeal.
A supporter who was with a group of Inter Milan fans died after being knocked down by a van amid violence before the match in Milan.
Four Napoli fans were hurt and one of them was stabbed, police said. Football fan dies amid clashes in Milan.
Inter's next three home league games are against Sassuolo on 19 January, Bologna on 3 February and Sampdoria on 17 February.
In addition to the punishment for Inter, Serie A have given Koulibaly a two-match ban after the Senegal international was sent off.
The 27-year-old received two yellow cards in a matter of seconds in the 80th minute. The first was for a foul on Matteo Politano, the second for sarcastically applauding the referee's decision.
Italian football federation president Gabriele Gravina said the racist chants and violence outside the stadium were "no longer tolerable".
"Football is the heritage of true supporters and as such should be defended from all those who use it as a tool to create tension," he said.
"We condemn all forms of both physical and verbal violence, with the aggravating circumstance of racial discrimination. We do not tolerate such behaviour ruining football."
Anti-discrimination organisation Fare commended former Chelsea boss Ancelotti for pledging to take his team off the field if such abuse happens again.
"Under Ancelotti's leadership the position of Napoli on racism and discriminatory abuse is clear, we note that the club previously threatened to take their players off the pitch before a match against Atlanta on 3 December," it said.
"Kalidou Koulibaly has consistently been the target of an alarming level of racist abuse while playing against clubs across Italy, including Atlanta, Juventus, Lazio and now Inter Milan.
"Too often, the punishments handed out by Italian football authorities are weak and ineffective."
After the match Koulibaly posted on Twitter: "I'm sorry about the defeat and especially to have let my brothers down.
"But I'm proud of the colour of my skin. To be French, Senegalese, Neapolitan, and a man."
The mayor of Milan, Guiseppe Sala, apologised to Koulibaly on his Facebook page, calling the abuse "a shameful act against a respected athlete".