The standard spelling of the noun meaning the state of being putrid or rotten is putrefaction, and this is the form that most commonly appears in all types of edited writing throughout the English-speaking world.
A few other forms are fairly common, however. Putrifaction, with an i in the second syllable, is an understable misspelling given the spelling of putrid. There are also putrefication, putrefecation, and putrifecation, which arise from a common secondary pronunciation (adding an extra syllable—pew-truh-fe-CAY-shun instead of pew-truh-FAC-shun) of the word. When you use any of these forms, some readers will see it as a misspelling. Putrefaction is always the safest choice.
Putrefaction came to English around the 14th century. It derives directly from a Middle French word with the same spelling and meaning. The French word, in turn, has origins in Latin, where putrefactio refers to the process of rotting.1 Although putrefaction and putrid share distant roots in Latin, they have come to modern English by different paths. Putrefaction does not take the i in putrid because its root has an e. Putrid has the i because its immediate origin, also in Middle French, is the adjective putride, with an i.2