12:42 PM PT -- Nigel Lythgoe tells TMZ ... “We live in a troubling time where a person is now automatically assumed to be guilty until proven innocent, a process that can take years. That is why, like Paula, I am glad to be able to put this behind me. I know the truth and that gives me great comfort.”

Paula Abdul says her sexual assault case against Nigel Lythgoe has ended, and "successfully" so -- the 2 former "American Idol" coworkers have reached a settlement.

According to court docs, they settled the case earlier this week and filed notice with L.A. County Superior Court. Their next step will be filing to dismiss the matter ... which had been scheduled to go to trial next year.

TMZ broke the story ... Paula sued Nigel last December, claiming he had sexually assaulted her twice -- once during their tenure at 'Idol,' when she alleged he had forcibly kissed and grabbed her genitals. She claimed the 2nd incident happened when they were doing "So You Think You Can Dance" together.

Nigel had denied all the allegations, and produced text messages and emails between the two -- which he said showed they had a very friendly and loving relationship for years ... even after the alleged incidents.

It's unclear if money exchanged hands as part of the settlement, but Paula's statement indicates she's satisfied with the terms. Paula tells TMZ ..."I am grateful this chapter has successfully come to a close and is now something I can now put behind me. This has been a long and hard-fought personal battle. I hope my experience can serve to inspire other women, facing similar struggles, to overcome their own challenges with dignity and respect, so that they too can turn the page and begin a new chapter of their lives."