Gemara

(Rav Ashi): Semichah is giving the title 'Rebbi' and permission to judge fines.

14a (R. Yehoshua ben Levi): Semichah does not apply in Chutz la'Aretz.

Suggestion: Fines cannot be judged at all in Chutz la'Aretz.

Rejection (Mishnah): We establish Sanhedriyos in Eretz Yisrael and Chutz la'Aretz.

Rather, he teaches that we do not give Semichah in Chutz la'Aretz.

Bava Kama 15b: Since half-damage is a fine, if a dog ate a (large) sheep, this is abnormal (a Toldah of Keren), so we do not collect it in Bavel (like all other fines).

84b (Rava): If a man or ox damaged an ox, we collect this in Bavel. If a man or ox damaged a man, we do not collect this in Bavel.

Question: We don't collect damages to a man, because the Torah says "Elohim (expert judges)." Only judges with Semichah may judge this (and we lack such judges in Bavel). Also regarding damages to an ox, it says "Elohim"!

Answer: We (judges in Chutz la'Aretz) are Sheluchim of ordained judges in Eretz Yisrael to collect damages to an ox, just like we are Sheluchim to judge cases of admissions and loans.

Question: Also regarding damages to a man, we should be Sheluchim to collect!

Answer: We are Sheluchim only to collect common things with Chisaron Kis (monetary loss). If a man hit a man, the victim loses money, but it is not common. Embarrassment is common, but there is no Chisaron Kis, so we do not judge it.

Contradiction: Rava said (Sof 84a) that we collect for an ox that damages an ox. He also said that we do not collect for an ox that damaged in Bavel.

If it struck a man, Rava should have taught that we do not collect even for a man who struck a man! Rather, it struck an ox, and we do not collect!

Answer #1: We do not collect from a Tam. We collect from a Mu'ad.

Objection: Rava taught that (becoming) Mu'ad does not apply in Bavel!

Answer #2: Regarding Shen and Regel, which are Mu'ad from the beginning, Rava taught that we collect for an ox that damaged an ox.

Gitin 88b: Rav Yosef was forcing a man to give a Get.

Question (Abaye): We (in Bavel) are commoners. We cannot coerce to divorce!

(Beraisa - R. Meir): "These are the judgments that you will put before them" (expert judges)", and not before commoners.

Answer (Rav Yosef): We are Sheluchim (to coerce Gitin) of the Semuchim in Eretz Yisrael, just like we are their Sheluchim to judge admissions and loans.

Question: If so, we should be Sheluchim also to judge thefts and injuries!

Answer: We are Sheluchim only for admissions and loans, for these are frequent.

Rishonim

Rif and Rosh (Bava Kama 30a and 8:2): If a man or ox damaged an ox, we collect this in Bavel, but not if a man or ox damaged a man, for the Torah says "Elohim". Judges in Chutz la'Aretz are Sheluchim of Semuchim in Eretz Yisrael to collect damages to an ox, just like they are Sheluchim for admissions and loans. They are Sheluchim only to collect common things in with Chisaron Kis. It is not common for a man to hit a man. Embarrassment has no Chisaron Kis. We collect for an ox that damaged an ox through Shen and Regel, which are always Mu'ad.

Rambam (Hilchos Sanhedrin 5:8): Three expert judges with Semichah in Eretz Yisrael are required for fines such as Gezelah, wounding, Kefel, four or five, rape and enticement and similar matters. Other monetary cases such as admissions and loans do not need experts. Even three commoners or one expert may judge them. Therefore, we judge admissions and loans and similar things in Chutz la'Aretz. Even though judges of Chutz la'Aretz are not called Elohim, they do Shelichus of Beis Din in Eretz Yisrael. They are not authorized to judge fines.

Rambam (9): Judges in Chutz la'Aretz judge only common matters with Chisaron Kis, e.g. admissions, loans, and property damage. They do not judge an animal that damaged another, for it is uncommon. They do not judge Kefel, for the victim did not lose. They don't judge fines that Chachamim fixed, e.g. for making a noise in Ploni's ear or slapping his face, or anyone who pays more than the damage or half-damage, except for half-damage of pebbles, which is Mamon, not a fine.

Or Some'ach (Hilchos Nizkei Mamon 2:9): It is unusual for a dog to kill sheep, so paying for the loss due to death is a fine (and we do not collect it in Bavel). However, it is normal to eat the meat, so one pays fully for this even in Bavel.

Rosh (Sanhedrin 1:1): The Gemara (32a) says that fines require interrogation. This is not limited to fines. The Gemara calls 'fines' anything that requires Mumchim. The Gemara (14a) says that fines were almost Batel, and it refers to everything that requires Mumchim. We concluded (Bava Kama 84b) that judges in Bavel are Sheluchim for everything common, just like admissions and loans. It says that wounds are not common. This implies that everything else, including theft, is common. Therefore, (in Bavel) we judge everything except for wounds; we do not need Mumchim, nor interrogation.

Tosfos (Gitin 88b DH b'Milsa): Even though there are no Semuchim in Eretz Yisrael nowadays, we are Sheluchim of previous Semuchim in Eretz Yisrael.

Poskim

Shulchan Aruch (CM 1:1): Nowadays judges judge admissions, loans, Kesuvos, inheritance, gifts and damaging another's property. These occur frequently, and a party loses money. Only experts with Semichah in Eretz Yisrael judge infrequent matters, even if there is Chisaron Kis, e.g. an animal that damaged or wounded another animal, or matters without Chisaron Kis even if they arise constantly, e.g. Kefel or fines that Chachamim imposed, e.g. for slapping someone.

Beis Yosef (DH u'Maskinim): We force men to divorce in some cases. Even though there is no Chisaron Kis, inability to remarrying is worse than a loss of money. The Ran says that accepting converts to bring them under the wings of the Shechinah is better than monetary cases, which are common.

SMA (3): Likewise, we judge all great matters.

Gra (1): Nowadays this is even in Eretz Yisrael (Tosfos Gitin 88b DH b'Milsa).

Tur: If an animal damaged an animal through Keren (goring) or its derivatives, we do not collect this in Bavel. If it is Tam, we hold that half-damage is a fine, which we do not judge in Chutz la'Aretz. We do not make an animal Mu'ad in Bavel. Even if it became Mu'ad in Eretz Yisrael and went to Bavel, we do not judge it, for it is not common.

Beis Yosef (DH u'Mah she'Chosav v'Chen, based on Rashi Bava Kama 84b DH Ein): Since we do not collect from a Tam, it cannot become Mu'ad.

Rejection (Tosfos Rid Bava Kama 84b DH v'Ha): We do not collect from a Tam of a deaf-mute, lunatic or child, yet we testify in front of an Apotropos to make it Mu'ad. We could do similarly for a Tam in Bavel! Rather, testimony to make a Mu'ad must be in front of Semuchim, which we lack in Bavel.

Taz (Sa'if 4 DH Kegon): The Tur explains that he has witnesses of an admission or loan. It cannot be that he wants to make him swear Heses without witnesses, for this is mid'Rabanan, so we would not say 'the Torah required Semuchim.'

Shulchan Aruch (5): This refers to written fines. If Chachamim come to impose new fines, we collect them everywhere.