The objective of this article is to explain the source from which we know that the universal message of the Torah, that is, the message for all nations of the world, is a message of action and meaning in life that includes seven general precepts.
The Rambam writes:
The First Man was commanded about six matters: idolatry, blasphemy, murder, forbidden relations, theft, and justice. Although all of these are traditions of Moses our teacher, and the intellect bows to them, it is clear from the words of the Torah that he was commanded about these things.
Laws of Kings, chapter 9, law 1
Furthermore, Noah was commanded not to eat any part of a living animal, as it is written (Genesis 9:4): "But you shall not eat flesh with its blood." These make seven commandments.
The source of this law is found in the Midrash with the following text:
""The Lord God commanded the man, saying, 'You shall surely eat of every tree of the garden'" (Genesis 2:16)
Bereshit Rabbah 16:6
""Command" (Vaitzav) refers to idolatry, as it is written (Hoshea 5:11): "For he began to go after the Tzav".
""God" refers to blasphemy, as it is written (Vayikra 24:16): "Whoever blasphemes the name of God shall die.".
""The Lord" refers to the judges, as it is written (Exodus 22:27): "The Lord, do not curse" (NT: referring to the judges).
""upon man" refers to murder, as it is written (Bereshit 9:6): "Who sheds the blood of man.".
""saying" refers to forbidden relationships, as it is written (Jeremiah 3:1): "A man must command his wife by saying.".
""You must eat from every tree in the garden" refers to theft (that is, you can only eat what I allowed you to eat).
It is interesting to note that the Talmud uses the same verse but with some minor changes:
Seven precepts were commanded to the sons of Noah: justice, blasphemy, idolatry, forbidden relations, murder, theft, and part of a living animal, as the verse says (Bereshit 2:16): "The Lord God commanded concerning man, saying: of every tree of the garden you shall eat.".
Sanhedrin 56b
""Command" (Vaitzav) refers to justice, as it is written (Bereshit 18:19): "I appreciate him (Abraham) because he instructs his children to do charity and justice.".
""God" refers to blasphemy, as it is written (Vayikra 24:16): "Whoever blasphemes the name of God shall die.".
""the Lord" (Elohim) refers to idolatry, as it is written (Exodus 20:2): "You shall have no other masters (Elohim).
""upon man" refers to murder, as it is written (Bereshit 9:6): "Who sheds the blood of man.".
""saying" refers to forbidden relationships, as it is written (Jeremiah 3:1): "A man must command his wife by saying.".
""from all the trees in the garden" and not from theft.
""You must eat" and not part of a live animal.
The commentators (see Torah Tmima Bereshit 2:16, points 38 and 39) indicate that learning the verse does not mean actually learning the seven commandments from the specific Torah text, because the simple meaning of the text is different. Rather, our sages received these seven commandments as a tradition (as Rambam says) and relied, as a reminder, on the verse from Bereshit to have an aid to remember what those seven commandments are.
I recommend this video to gain a broader perspective on what the precepts are and, in addition, to consider a phrase from the Talmud that indicates there are 30 precepts:
These are the thirty precepts that the sons of Noah received (explaining Zechariah 11:12)
Julin 92a