Can sewage smell penetrate indoors because of inadequate venting?

I have a basement toilet that is connected to a 4" buried sewer pipe but its vent that goes three stories up to the roof is only 2". The toilet seems to have a permanent smell of urine which starts even very soon after I clean it with Ajax and before anyone has used it. I understand the toilet has its own P or S trapping module intended to prevent backflow of gases from the pipe but could it be that the vent inadequately forces air out and some of it is finding its way indoors through the 1-2 sec period of flushing?

This is a hypothesis and I have no other explanation. My other toilets that are right next to a different, 4" vent stack don't smell and it has nothing to do with the cleaning or sanitation treatment. Moreover, I have a sewer access point (for snaking and cleanouts) above the toilet behind the wall and when I open the plug, it has the same smell as what collects inside the bowl.