I think It is: The Abyss.

Because the radiaton, and the sea singnals. Besides the URSS flag.

In the movie an American nuclear Submarine has to be rescued before URSS recovered its nuclear warheads.

But at the end of the movie we realised that there is an alien ship involved. What explained the flyng saucer signal next to the toilet.

The ring into the toilet... I barely remember... I think that The Main character played by Ed Harris throwed away his ring on the toilet.