Oh man, you have much to learn homie.
Don't get my wrong, a lot of people don't know this, it's okay. ^_^
First, I must correct the spelling, it's Shawn Carter.
Second, Jay-Z did produce it, that is correct, as did Shawn Carter.
Shawn 'S-Dot' 'Jay-Z' Carter.
After he retired, he took back his given name, Shawn Carter.
S-Dot came from him being in buisiness, he would sign his name, S. Carter, hence, S-Dot Carter.
Also another example of this, is when Mike does his shout-outs at the end of 'We Major', he says, "I gotta say what up yo my executive producer, Mr. S-Dot Carter."
Hope that helped you.