And “Peace be upon Noah among the worlds” is a quoted sentence. Al-Zamakhsharī said:
“We left for him among later nations this expression—‘Peace be upon Noah’—meaning that they invoke peace upon him and pray for him. This is quoted speech, as when you say, ‘I recited: “A surah which We have sent down.”’”
Fifth: He says, “Peace be upon Noah among the worlds.” He, glorified be He, thus states that this peace upon Noah is among the worlds. It is understood that this peace among them consists of their invoking peace upon him: they all invoke peace upon him, praise him, and pray for him. He thus mentions peace being invoked upon Noah among them.
As for Allah’s own bestowal of peace upon him, it is not restricted to them. For this reason, it is not prescribed to ask Allah Most High for this particular form of blessing. One does not say, “Peace be upon the Messenger of Allah among the worlds,” or, “O Allah, bestow peace upon Your Messenger among the worlds.” If this were Allah’s own bestowal of peace, it would be prescribed to ask Him for it in the same manner in which He bestowed it.
As for their statement that Allah bestowed peace upon him among the worlds and left it for him among later generations:
Allah, glorified and exalted be He, has preserved for His prophets and messengers invocations of peace and favourable praise among those who came after them, as a reward for their patience, their conveyance of their Lord’s messages, and their endurance, for Allah’s sake, of the harm inflicted by their peoples. He states that what has been left for Noah extends throughout the worlds, and that this greeting is established among them all, with none devoid of it. He has perpetuated it for him among the angels, humankind, and jinn, generation after generation and world after world, rewarding Noah, peace be upon him, for his patience, his fulfilment of his Lord’s rights, and his being the first messenger whom He sent to the people of the earth. Every messenger after him was sent with his religion, as Allah Most High says:
“He has ordained for you of religion what He enjoined upon Noah” (42:13), to the end of the verse.
As for their claim that this is the statement of Ibn ʿAbbās, it has already been explained that Ibn ʿAbbās and others meant only that invoking peace upon them is a form of favourable praise and honourable remembrance. They were describing the meaning and benefit of invoking peace upon him. And Allah, glorified be He, knows best.