The Most Important Son of Noah.
—In Rabbinical Literature:
Although Shem is unanimously declared by the Rabbis to have been the youngest son of Noah (comp. Japheth in Rabbinical Literature), yet he is always named first, being the most important of the three brothers. Indeed, he was born circumcised; he was the ancestor of Abraham, Isaac, and Jacob; he was priest and prophet; and he was one of the eight righteous who are mentioned twice in Gen. xi. 10 and who were allotted a portion both in this world and in the world to come (Sanh. 69b; Tan., Yelammedenu, Noaḥ; Midr. ha-Gadol on Gen. ix. 18, xi. 10, ed. Schechter, cols. 142, 186). Shem is styled “the’ great one” (“Shem rabba”; Sanh. 108b). According to Gen. R. xxx. 6, it was Shem who offered the sacrifices on the altar after Noah came out of the ark (comp. Gen. viii. 20), as the latter, having been crippled by the lion (see Noah in Rabbinical Literature), was unfit for the priestly office. Noah gave to Shem the priestly garments which he had inherited from Adam (Num. R. iv. 6). Shem is extolled by the Rabbis for his filial devotion in covering his father’s nakedness (Gen. ix. 23). Although his brother Japheth assisted in this praiseworthy act, it was Shem who suggested and began it, his brother not arriving on the scene until Shem was already on his way with the garment. Therefore Noah, in blessing these two sons (ib. verse 27), declared, so the Rabbis think, that the Shekinah was to dwell only in the tents of Shem (Yoma 10a; Tan., Noaḥ, 21; Gen. R. xxxvii. 9; comp. Jubilees, vii. 9, where it is said that the garment was Shem’s). Shem’s reward for this deed is seen in the fact that the Jews, his descendants, cover themselves with the ṭallit and phylacteries, and remained untouched when the Assyrians, who also were descendants of Shem, were destroyed by an angel in the time of Hezekiah (Tan., Yelammedenu, l.c.; Ex. R. xviii. 5).

Legends.
The Rabbis identify Shem with Melchizedek, King of Salem, who is termed “a priest of the Most High,” and who came to meet Abraham after the latter had defeated the four kings led by Chedorlaomer (Gen. xiv. 18-20). According to this account, Shem, as a priest, came to Jerusalem (with which Salem is identified by the Rabbis), of which city he became king, it being the proper place for the establishment of the cult of Yhwh. He went to meet Abraham to show him that he was not angry with him for having killed the Elamites, his descendants (Midr. Agadah on Gen. l.c.). Shem, however, forfeited the priesthood by mentioning in his blessing Abraham’s name before that of God, so that God took his office from him and gave it to Abraham (Ned. 32b; Pirḳe R. El. xxvii.). According to the Midrash Agadah (l.c.) Shem himself asked God togive the priesthood to Abraham, as he, in his prophetic capacity, knew that he (Shem) would have no children eligible for the sacerdotal office. Contrary to the Pirḳe R. El. and Gen. R. (xliii. 10), the Midrash Agadah explains that it was Shem who gave tithes to Abraham, showing that he recognized him as priest (see Gen. R. xliii. 7). The Rabbis point out that in certain cases Shem ranked as the equal of Abraham; so that the latter was afraid lest Shem might be angry at him for having slain the Elamites and might curse him (Gen. R. xliv. 8; Tan., Lek Leka, 19). In another instance God made a compromise between Shem and Abraham, namely, with regard to the name of the Holy City, the place of the Temple, which Abraham had called “Jireh” (Gen. xxii. 14; see Jehovah-jireh) and which Shem had called “Salem.” God united both names; and thus arose the name “Jerusalem” (Gen. R. lvi. 16).

Shem is supposed by the Rabbis to have established a school (“bet ha-midrash”) in which the Torah was studied, and among the pupils of which was Jacob. Later, Shem was joined by Eber; and the school was called after both of them. Besides, the school was the seat of a regular bet din which promulgated the laws current in those times. Thus Esau was afraid to kill Jacob, lest he should be condemned by the bet din of Shem and Eber. The bet din of Shem proclaimed the prohibition of and the punishment for adultery; and according to this law Judah condemned Tamar to be burned (‘Ab. Zarah 36b; Gen. R. lxiii. 7, lxvii. 8). Shem’s bet din was one of the three in which the presence of the Shekinah was manifested (Mak. 23b). At Abraham’s death Shem and Eber marched before his bier; and they indicated the place that was suitable for his burial (Gen. R. lxii. 6, according to the emendation of the text in Yalḳ., Gen. 110). At the division of the earth among the three sons of Noah, Shem’s lot consisted of twenty-six countries, thirty-three islands, twenty-six out of seventy-two languages, and six out of sixteen scripts. Thus Shem took one script more than either of his two brothers: and this was the Hebrew script, in which the Torah was written. The other five were Egyptian, Libyan, Assyrian, Chaldean, and Guṭazaki (Guzarati ?) (Midr. ha-Gadol on Gen. x. 32, col. 182).

—Critical View:
Shem is not an individual, in the sense that one person by that name came forth with his father and brothers from the ark, and had a share in the scene described in Gen. ix. 18-27. Neither does the name in itself suggest geographical or racial entities. It recalls more probably some ethnic deity that had become the “heros eponymus” of his worshipers. As it now occurs, the name has no theophorous character; but it has been suggested that “Shem” must be considered a corruption or abbreviation of a name similar to Shemu’el (see Samuel), the element “Shem” meaning “son” in the combination. This suggestion—though none of the critics seems to have noticed it—receives a strong degree of probability from the blessing spoken over Shem (ib. verse 26). There is no doubt that the pointing of the text is incorrect. Budde proposes to omit the (which Grätz would read “ohole” = “tents”), and then vocalize: “Beruk Yhwh Shem” = “Shem is blessed of Yhwh.” This would at once place this “blessing” in the category, so numerously represented in Genesis, of name oracles. From the oracle the name is readily reconstructed as “Shemaiah” or “Shemu’el,” the “Elohe Shem” in the text indicating the latter possibility.

These oracles are always the primary elements from which the legend in which they are found embedded is a development. That Japheth also originally had a theophorous form is indicated in the oracle spoken concerning him (Gen. ix. 27; comp. the name ). It is plain that Canaan should not appear in this group. Ham is the brother of Shem; and it was he who committed the unseemly deed. The substitution of Canaan for Ham is secondary. The curse upon him (Canaan) displays the temper of the centuries when Yhwh and Baal were struggling for the ascendency (see Elijah). As Shem represents Yhwh, he is proclaimed the master, while Canaan is doomed to servitude. As Israel is the people of Yhwh, Shem(yahu), i.e., “the son [of Yhwh],” naturally must be Israel’s progenitor. In substance this is also the explanation of those scholars who reject the suggestion that “Shem” is a name like “Shemu’el.” They read into “Shem” the signification of “prominence,” “mastership.” The people descended from Shem is thus the master people destined to “lord it” over Canaan, the slave people committing such dire atrocities as are hidden in the legend of Noah’s exposure. According to Budde, Japheth—which name means “beauty”—represents the Phenicians, while Canaan, signifying “lowness,” “vulgarity,” represents the aboriginal population of Palestine. Thus this triad would result: lordship (Shem), beauty (Japheth), and meanness (Canaan).

The Zohar has something to say about Melchizedek as well:

Soncino Zohar, Bereshith, Section 1, Page 87a – AND MELCHIZEDEK KING OF SALEM BROUGHT FORTH BREAD AND WINE. R. Simeon adduced here the text “In Salem also is his tabernacle” (Ps. LXXVI, 3). He said: ‘When God decided to create the world, He first produced a flame of a scintillating lamp. He blew spark against spark, causing darkness and fire, and produced from the recesses of the abyss a certain drop which He joined with the flame, and from the two He created the world. The flame ascended and encircled itself with the Left, and the drop ascended and encircled itself with the Right. They then crossed and changed places, going up and down alternately until they were closely interlocked, and there issued from between them a full wind. Then those two sides were made one, and the wind was set between them and they were entwined with one another, and so there was harmony above and harmony below; the grade was firmly established, the letter he was crowned with vau and vau with he, and so he ascended and was joined in a perfect bond. This is alluded to in the words “Melchizedek (lit. king of righteousness) king of Salem” (lit. completeness), i.e. the king who rules with complete sovereignty. When is he completely king? On the Day of Atonement, when all faces are illumined. According to another explanation, “Melchizedek” alludes to the lower world, and “king of Salem” to the upper world; and the verse indicates that both are intertwined inseparably, two worlds like one, so that the lower world also is the whole, and the whole is one. “Brought forth bread and wine”: signifying that both of these are in it. AND HE WAS PRIEST OF GOD MOST HIGH: i.e. one world ministers to the other. “Priest” refers to the Right, and “Most High God,’ to the upper world; and hence a priest is required to bless the world. For this lower world receives blessings when it is associated with a High Priest; hence there is a special force in the words “and he blessed him and said, Blessed is Abram to the Most High God”. After this model it behoves the priest on earth to intertwine his fingers when blessing in the synagogue in order that he may be linked with the Right and that the two worlds may be linked together. BLESSED IS ABRAM. The words of the text are a prototype of the formula of blessing (used by the Israclites). “Blessed is Abram” (in the sense we have given to it) corresponds to “blessed art Thou “. “To the Most High God” corresponds to “O Lord our God”. “Possessor of heaven and earth” corresponds to “king of the universe. Further, AND HE BLESSED HIM indicates the course of blessing from below to above; BLESSED IS THE MOST HIGH GOD indicates from above to below. AND HE GAVE HIM A TENTH OF ALL: so that he should cleave to the place where the link was formed with the lower world.’

As they were going along they came across R. Yesa and a certain Judean with him who was explaining the text “To David: Unto thee, O Lord, do I lift up my soul” (Ps. XXV, 1). He said: ‘Why is the inscription of this psalm simply “to David” and not “A Psalm of David”? It is because the real meaning is “for the sake of David”, i.e. of his grade. “Unto thee, O Lord”, i.e. upward-striving; “my soul”, i.e. David himself, his original grade; “I lift up”: to wit, I cause to ascend, since David was ever striving to rise to a higher grade and to link himself to it firmly. Similarly it was for the sake of his grade that David uttered the words “To David: Bless the Lord, O my soul” (where the word eth indicates his desire to be linked above) “and all that is within me bless his holy name” (Ps. CIII, 1), referring to the “beasts of the field” which are called “inwards”.’ Said R. Eleazar to R. Yesa, ‘I see that you have come in company with the Shekinah.’ He said, “Assuredly it is so. I have been walking with him three parasangs, and he has told me ever so many excellent things. I hired him as a porter, not knowing that he was the shining light which I have discovered him to be.’ R. Eleazar then said to the man, ‘What is your name?’ He said: ‘Joezer’. Whereupon he said: ‘Let Joezer and Eleazar sit together.’ So they sat down on a rock in that field. The Judean then commenced to discourse on the text I, even I, am he that blotteth out thy transgressions, for mine own sake, and thy sins I will not remember (Is. XLIII, 25). He said: ‘The word “I” occurs here twice: once in reference to Sinai (cf. “I am the Lord thy God”, Ex. XX, 2), and the other in reference to the creation of the world (cf. “I have made the earth and created man upon it”, Is. XLV, 12), to show that there is no division between the upper and lower worlds. “That blotteth out thy transgressions”: not merely removing them, so that they shall never be seen more. “For mine own sake”: i.e. for the sake of the mercy which I dispense, as it is written, “For the Lord thy God is a merciful God” (Deut. IV, 31). Another explanation of the words”that blotteth out thy transgressions for mine own sake” is as follows. Sinners in this world impair the influence of the upper world, for when they sin, mercy and the supernal light depart, and the stream of blessing does not descend to this world, and this grade (of mercy) does not take up the blessings from above in order to convey them to the lower world. Hence God acts “for His own sake”, in order that the stream of blessing should not be withheld. Similarly it is written, “See now that I, I am he” (Deut. XXXII, 39), to show that there is no division between the upper and the lower. See now, in this way, when there are righteous men in the world, blessings are sent to all worlds. When Abram came, blessings were sent to the world, as it is written, “And I shall bless thee, and be thou a blessing, i.e. that blessing should be found both above and below for his sake. When Isaac came he taught the world that there is a judge executing judgement above to punish the wicked, and he invoked justice upon the world in order that its inhabitants might fear God. When Jacob came he obtained mercy for the world and perfected men’s faith in God. Hence in the days of Abram MELCHIZEDEK KING OF SALEM (salem=completeness), i.e. God whose throne was then established in its place and whose sovereignty therefore became complete, BROUGHT OUT BREAD AND WINE, i.e. produced the appropriate food for the whole world, and did not withhold blessing from all the worlds; from the upper grades He brought forth food and blessings for all the worlds. AND HE WAS A PRIEST TO THE MOST HIGH GOD, the whole thus being in the most perfect order; to show that as the wicked upset the world and cause blessing to be withheld, so the righteous bring blessing to the world and for their sakes all its inhabitants are blessed. AND HE GAVE HIM A TENTH OF ALL, to wit, of those blessings which issue from “all”, the source of all the blessings which descend upon the world.

The subject of the Melchizedek priesthood will be covered in greater detail in our study on the book of Hebrews beginning later this year. An excellent study on Hebrews It provides information such as this:

Sha’ul identifies Yeshua with the Melchizadek figure in Ps. 110:4. While the Talmud identifies Melchizadek as a son of Shem (b.Nedarim 32b) the Qumran community had a more mystical view. In a document found among the Dead Sea Scrolls (11Q13) Melchizadek is presented as a future Messianic figure who would redeem those that are his. This Melchizadek is called in the Qumran document “El” and “Elohim,” moreover Is. 61:1-2 is quoted in reference to him with “Melchizadek” substituted for YHWH. The redemption of this Melchizadek figure is also tied, in the Qumran text to the Day of Atonement. It is also significant that the Qumran community believed in two Messiah’s “the Messiahs of Aaron and Israel” (1Qs 9:10-11) A priestly Messiah and a kingly Messiah. 

proponent himself, who observed that the expression does not occur in the oldest exemplar of 1QS: 4QSe breaks off after 8:16 and only resumes at 9:12.51 Thus supplied with the necessary “jot” on which to build their case, those scholars who already saw a plural messianism in the scrolls lost little time in making the most of the serendipitous reading. Kuhn claimed that final proof of medieval tampering with CD had finally been supplied by 1QS.52 It was even suggested by Burrows and Milik that we now have at Qumran three “messianic” figures, a prophet and the two Messiahs.53 They rejected the supposition of Brownlee that the passage indicates that the prophet was the Messiah, and that the two “anointed ones” were his followers.54 “This involves,” says Burrows, “the decidedly precarious assumption that the Messiah is one who is not so designated, while those who actually are so designated are only his followers.”55 So, adds Milik, the plural messiahship implied by the term mešîahiê of Aaron and Israel is “grammatically . . . the only possible interpretation.”