Hi I am wondering what the correct term is for play a sequence of notes where you leave all the notes ringing together. I'm not talking about rolled chords, I'm talking about having a larger amount of time between the attacked notes than that.
Uhm, arpeggio? "Leave all the notes ringing together" (the effect depending on the sustain of the respective instrument, so one pretty much needs a percussive instrument) is rather the definition of arpeggio. Otherwise you have something else, like a broken chord.
You cannot arpeggiate on a monophonic instrument (like voice or most wind instruments).
From the perspective of piano, a measure that has a clear arpeggiation may or may not have an indication to use the pedal, which would pretty much leave every note to ring until it was released. Not pedaling would, in turn, have a more staccato effect. In my experience, it's either called "pedalling," which is fairly straightforward, but which only relates to piano.
The better series of terms to differentiate how notes are sounded (across any instrument) would be 1. staccato, 2. marcato, and 3. legato, for 1. short, 2. "normal," and 3. "flowing into one another," respectively. To my knowledge there is no specific term when an arpeggio or any other named phrase is called to be one of these three, so I'd merely put what you're talking about at "an arpeggio played legato."