[verb ない-form 〜ない → 〜] + ずに する → せずに
ずに means "without doing": the main action happens, and the one before ずに is left undone. Take the ない-form of a verb and replace ない with ずに – 食べない gives たべ食べずに, 持たない gives もた持たずに. Both halves of the sentence have the same subject.
する becomes せずに, and くる来る becomes こ来ずに, read こずに.These go beyond "without doing".
With two actions where one replaces the other, ずに is "instead of".
In writing, ずに often drops the に. The meaning is the same and the tone more formal.