Nope, ringing is the effect of a high-pass (or band-pass) filter - it's the opposite of smoothing.

JPEG looses information by quantization, not low-passing. It is true that small high frequency coefficients will get rounded to 0 during quantization, and this does save quite a few bits. But the ringing is just from being imprecise about each DCT coefficient, coding a value that is either too large or too small.