What are the connotations of "u've'Chol asher Amarti Aleichem Tishameru"?
Rashi #1: It adds a Lo Sa'aseh to every Asei. 1
Rashi #2: The Torah is teaching us here that idolatry (mentioned right after this) is on a par with all other Mitzvos combined. 2
Ramban: With reference to the following phrase, the Torah is warning us here to strictly adhere to all the warnings concerning Avodah-Zarah, 3 and that moreover, one may not even mention 4 their names 5 either personally, or cause them to be mentioned by their adherents.
Seforno: With reference to the following phrase, the Torah is warning us here although one needs to take care not to transgress any of the Mitzvos of the Torah, 6 when it comes to Avodah-Zarah, one needs to take additional care even not to mention it by name.
Rashbam #1: The Torah is reiterating here, all the Mitzvos that it presented since the beginning of the Aseres ha'Dibros until now.
Rashbam #2 (citing Shabbos, 18a): It is coming to add Shevisas Keilim (not allowing our vessels to work for us on Shabbos). 7
Da'as Zekenim (12) #1: Even though I commanded you about [not afflicting] a Ger, do not join with him too much.
Da'as Zekenim (12) #2: After this, the Torah commands about the Yamim Tovim, i.e. times of Simchah. One is prone to speak improperly amidst wine or intoxication - therefore, it says 'guard everything I commanded!' Know Hashem in all your ways!
Mechilta: It is coming to add a prohibition against performing acts of Sh'vus on Shabbos. 8
Based on the principle "Hishamer", "Pen" and "Al" are Lo Sa'asehs. See Sifsei Chachamim and Ramban. Citing Eruvin, 96a, the Ramban concludes that 'Hishamer of an Asei is an Asei'. See Rashi, Devarim 13:1.
Rashi: If someone transgresses it, it is as if he transgressed all the Mitzvos, and conversely for someone who is careful not to transgress. See also Ba'al ha'Turim.
Not to worship them or bow down to them, to sentence to death those who worship them, not to make a like image or a picture of them. (Ramban).
The prohibition is confined to mentioning them to praise them, but with regards to mentioning them derogatively, the Torah writes in Eikev, Devarim, 7:26 "Shakeitz Teshatzenu ... ".
Ramban: Such as K'mosh, god of Mo'av, Malkom, god of Amon and Asima, god of Chamos - even without mentioning the word 'god'. (Why does the Ramban list idolatries mentioned in Tanach? Sanhedrin 63b permits mentioning them? - PF). The Gemara only permits it nowadays, because the idols mentioned in T'nach are obsolete, whereas the Ramban is commenting on the Pasuk, which is speaking to Yisrael at a time when they were still being worshipped.
This also appears to be the way Targum Onkelos and Targum Yonasan explain the Pasuk.
See Shabbos, 18a.
See Torah Temimah, note 85. This is probably an Asmachta
What does "ve'Shem Elohim Acherim Lo Sazkiru" mean?
Rashi #1: It is a prohibition against mentioning the name of an idol, even as a landmark. 1
Ramban: Refer to 23:13:1:2.
Seforno: Refer to, 23:13:1:3.
What are the connotations of "Lo Yishama al Picha"?
Rashi and Ramban (citing Sanhedrin, 63b) #1: It is a prohibition against entering into a partnership with a Nochri, thereby causing him to swear by the name of his god, should a disagreement between the partners occur. 1
Ramban #2: One may not enter into a partnership with a Nochri in case one mentions the name of one's partner's god, to say to him, 'Do me a favor in the name of your god!', or in some other way. 2
Seforno: It is a prohibition against giving one's consent to a Nochri to mention (or to swearby) his god.
Sanhedrin, 63b: It is an Azharah (a warning) against being a Madi'ach. 3
Sanhedrin, 63b:It is a prohibition against making a Neder or a Shevu'ah in the name of an Avodah Zarah or causing others to do so.
Moshav Zekenim: How do we accept oaths from Nochrim? Perhaps we explain "Tazkiru" to be that the Yisrael mentions its name - 'by your god, do me a favor.' (The Rema (OC 156:1) says thatsome are lenient nowadays because Nochrim intend for the Creator, and not for idolatry.)
Why does the Torah jixtapose not mentioning the name of a god to the Mitzvah of Shabbos?
Oznayim la'Torah: Because keeping Shabbos and learning Torah on Shabbos (when one is free from work) is a direct negation of Avodah Zarah, inasmuch as it expresses one's belief that Hashem created the world in six days and rested on thr seventh. 1
Oznayim la'Torah: Hence the Gemara states in Shabbos, 119 that if someone keeps Shabbos according to the Halachah, then, even if he worshipped Avodah Zarah lie the generation of Enosh, Hashem will pardon him