on what the Buddha taught
Ariyasāvaka is often found translated as noble disciple.
It is a compound formed with the words ariya and sāvaka. Ariya is said to refer to the generally approved and esteemed customs and ideals of the Aryan clans, covering the racial, social and ethical aspects translated as noble and sāvaka as disciple.
Noble disciple seems to indicate the disciple as the noble one, but note that for an arahant 'sāvaka' is used, not ariyasāvaka, which makes this a little odd. It ironically also seem to make some people wonder who exactly is when noble.
Another rendering is disciple of the noble one(s). At least here people can be disciples without worrying about their status as nobility, but again for the arahant sāvakā is used. Thus if for the arahant noble falls off then it is a bit odd to refer to them as noble ones.
Both translations leave some oddness by trying to assign 'noble' to a certain person. But noble here stood for the social and ethical direction. So it is about the learning/following the good, the highest, righteousness, of all of those traits for which aryan stood example. And for this generalization we can use nobleness instead. A monastic is not a noble one, a virtous one is not a noble one, the exalted one is not a noble one, and the dhamma is not noble (after all there are unwholesome dhammā) but nobleness is and noble principles are.
So then, disciple of nobleness? But what exactly would that mean? The word disciple implies a certain relationship with nobleness and since nobleness is an abstract word it makes this to ambiguous. Then lets look at sāvaka as well. The dictionary says:
Sāvaka [fr. śru] a hearer, disciple.
And with hearer we don't have the problem as with disciple. It also has a long standing relation with oral traditions so that seems to fit. Then for ariyasāvaka we have 'hearer of nobleness' (as in hearer of the good). Even when the exalted ones' words are long gone, one can set certain things as nobler, higher, not to admire but to advance to. Advancing this way keeps siding with the good. And when the dhamma is discovered then it is set as the new highest. So as principle it seems to fit as well.
The ariyasāvaka and arahant are both sāvakā (hearers; hearers of the exalted one). Since the arahant is done with going, siding, that direction towards the good does not apply. This way it can be understood why an arahant is then not an ariyasāvaka.
Ariyasāvaka is a compound of which we made the two part clear. We have removed an assumed person from being assigned to as noble and we have removed an assumed person from being assigned to having disciples. And this has nothing to do with regarding someone as noble or regarding someone having disciples, it is just that this is not what this is here about. Now noble is inclined for nobleness; again it is not a person who is noble but rather the nobleness of what is noble. And of what is noble that a hearer get to be educated about. As compound 'hearer of what is noble' fits grammatically as well, a bahubbīhi compound may require the addition of a pronoun. So now with the problems gone and making sense both grammatically and in meaning, we can use either 'hearer of nobleness' or 'hearer of what is noble'. Both say the same but perhaps the least abstract version makes it even more clear. Then, for example, a line could go something like this:
Now there, almsmen, having learned, the hearer of what is noble pays attention well...
Ariya (adj. — n.) noble.
— sāvaka hearer of what is noble.