How to increase public loyalty?

[Omanes Alexandrapolites]

how can i remedy corruption?

Buildings which boost law, are your resolution to corruption! They may never completely destroy this horrible R:TW statistic, but they still can drastically reduce it under most circumstances. Also moving the capital closer to your corrupt settlements can also reduce the statistic, but may, if not centralised, inspire others to become corrupt further away.

oh, what would happen if the the Senate (i forgot the name of the office that does it) actually checks your accounting for correctness (they threatened to do that to me lol)?

It's just a simply fine of either 5,000 or 10,000 denarii. According to the text files it can't go any higher than that. However, the text files aren't always perfectly accurate sources, so, unfortunately, I may be wrong.

However, since the not-so-wise senate is now challenging your supremacy in this way, it may be a good idea to fortify your borders against the other Romans, should the senate demand your faction leader commits suicide and you wish to tell them no (resulting in exile and civil war - the other Romans always support the senate).

Hope this helps you, cheers!