The noun integrity means to be honest and having strong moral principles as well as being completely undivided.  

Synonyms are solidarity, honesty, sincerity or unimpaired.

The word origins from Old French (1400) integrité meaning “innocence, blamelessness, chastity, purity”. It relates to the Latin word integer “whole” in the sense of wholeness or perfect condition.

 Golf prides itself most on its moral integrity.

Tom is a man of integrity.

The teacher would prefer students showing integrity and failing an exam instead of cheating and passing with wrongdoing.