Gemara

(Ravina): If one is afraid and does not know why, he should say "Shema Yisrael..." If there is smelly filth in the area (he may not say Shema), he should say 'the goat at the butchery is fatter than I.'

Berachos 22b (Mishnah): One who prays must distance himself four Amos from smelly water, water in which flax was soaked, and from excrement.

(Beraisa #1): If during Shemoneh Esreh one saw excrement in front of him, he should go four Amos in front of it.

Contradiction (Beraisa #2): He should go (four Amos) to the side.

Resolution: If he can, he goes in front of it. If not, he goes to the side.

25a (Rav Huna): If there is a Re'ach Ra she'Yesh Lo Ikar (a bad smell with a source), one must distance himself four Amos to say Shema;

(Rav Chisda): He distances four Amos from where the smell ends.

Support (for Rav Chisda - Beraisa #1): One may not say Shema or Shemoneh Esre facing excrement of people, dogs, pigs, or chickens, or a foul smelling wasteheap. If it was ten Tefachim above or below him, he may say Shema next to it. If not, he must distance himself until he cannot see it. If there is Re'ach Ra she'Yesh Lo Ikar, he must distance himself four Amos from the smell.

(Rava): The Halachah is unlike this, rather, it is like Beraisa #2:

(Beraisa #2): One may not say Shema facing human excrement, nor that of dogs or pigs if hide was put in (to be tanned).

Question: What is the law of Re'ach Ra she'Ein Lo Ikar?

Answer (Rav Sheshes): Some Talmidim learn while others sleep and pass gas. One may learn Torah (when there is a smell); one may not say Shema.

Rishonim

Rambam (Hilchos Kri'as Shema 3:8): One distances four Amos from excrement or urine to say Kri'as Shema. This is if it is in back of him or to the side. If it is in front of him, he distances until he cannot see it.

Kesef Mishneh: The Rambam holds that the sides are like in back of him. R. Mano'ach says that the sides should be like in front of him, like we say about a bent Lulav. He suggests that the Rambam means that when one cannot arrange that it is in back of him, the sides are like in back, but when it is possible, they are like in front of him, like the Mishnah (22b).

Rambam (12): One distances four Amos from Re'ach Ra she'Yesh Lo Ikar, and reads Kri'as Shema, if the smell ended. If he can smell it, he distances to where he cannot smell it.

Rebuttal (Ra'avad): He must distance four Amos from where the smell ends. The Beraisa that supports Rav Chisda says so. Rava did not say that the Halachah does not follow this Beraisa regarding this, rather, regarding excrement of dogs and pigs. Also Rav Hai Gaon says so.

Kesef Mishneh: R. Mano'ach says that we discuss covered excrement in back of him or to the side. If it is in front, he must distance until he cannot see it.

Rosh (Berachos 3:46): Some say that just like a Hefsek (interruption, i.e. he is 10 Tefachim above or below the excrement) helps for excrement, it helps for a foul smell. Rabbeinu Moshe says that since he can smell it, the Hefsek does not help at all. When one must distance himself so he cannot see it, at night he must distance so far that he could not see it if it were day. The Yerushalmi says so. However, if he is 10 Tefachim above or below, even if he sees it, since he is in a separate place, this is called "your encampment is Kodesh." Based on this, if excrement is in front of the door, one may read Shema in the house, even if he sees it and even if he is within four Amos. If he cannot smell it, he is in a different Reshus. It is as if the door is locked. It is as if the excrement is covered by glass, so it is permitted. (When one is in the same Reshus) he must distance so the excrement is four Amos in back of him, or so far in front of him that he cannot see it, even if he cannot smell. The Halachah follows Rav Chisda, for Beraisa #1 supports him. Even though the Halachah does not follow Beraisa #1 regarding excrement of dogs and pigs, the Halachah follows Beraisa #1 regarding Re'ach Ra she'Yesh Lo Ikar, for Beraisa #2 does not argue about this.

Poskim

Shulchan Aruch (OC 79:2): If human excrement was in back of him, one must distance four Amos from where the smell ends. Even if he has a sickness and he cannot smell, he must distance four Amos from where one who can smell could smell it.

Beis Yosef (DH Re'ach): Rav Hai Gaon, the Ra'avad, R. Yonah and the Rashba hold like the Rosh, that the Halachah follows Rav Chisda. The Rambam disagrees. It seems that the Rambam holds that since we partially rejected the Beraisa that supports Rav Chisda, it is as if it is erroneous, and even the part that was not explicitly rejected does not support him. We revert to the rule that the Halachah does not follow a Talmid against his Rebbi; Rav Huna was Rav Chisda's Rebbi. Migdal Oz says that the Rambam rules like Rav Chisda. His text said that Rav Chisda forbids until where the smell ceases. The Rif brought the Beraisa supporting Rav Chisda, but says that the Halachah is unlike this, rather, it is like Beraisa #2. It seems that he holds that the Halachah follows Rav Chisda, for if not, why did he bring Beraisa #1? He should bring only Beraisa #2! Since the Rif, Rosh R. Yonah and Rashba agree, we follow them.

Mishnah Berurah (2): We require "your Machaneh is Kadosh". Chachamim understood that one's Machaneh is four Amos, so there may not be a bad smell within his four Amos.

Prishah (1): The Rashba says that exposed excrement is a Re'ach Ra she'Yesh Lo Ikar. The Beis Yosef said that this is obvious. This is because if it is covered, one need not distance past where one cannot smell it. It is like Re'ach Ra she'Ein Lo Ikar. A proof is one who passed gas. The smell comes from excrement, but since the excrement is covered, it is considered Ein Lo IKar.

Kaf ha'Chayim (1): The Roke'ach says that if a smell exudes from covered excrement, one must distance four Amos from where the smell ends. The Kesef Mishneh, Rashi and Rambam connote that Re'ach Ra she'Ein Lo Ikar is only gas that left the body. It seems that the Shulchan Aruch agrees, for he did not say that covered excrement is Ein Lo Ikar. He mentioned only gas. When possible, one should be stringent. If he cannot, one may rely on the Rashba and Prishah that it suffices to go to where he cannot smell it.

Kaf ha'Chayim (2): We distance from every Re'ach Ra she'Yesh Lo Ikar like from excrement . Even one who cannot smell must distance from a smell that bothers people. However, we permit something that naturally has a bad smell not due to rotting, e.g. kerosene. Smelling a Neveilah or other bad smell damages one's Nefesh. (A proof is the spices we smell on Motza'ei Shabbos to restore the Nefesh.) Also this applies only to something rotting.

Kaf ha'Chayim (3): If one prayed within four Amos from where the smell ends, he was Yotzei, for this law is only mid'Rabanan. If he said Shema within four Amos, he repeats Shema without the Berachos.

Kaf ha'Chayim (8): The Mechaber did not need to write the law of a blind person, for we may learn from saying Shema at night.

Shulchan Aruch (ibid): If it is in front of him must distance as far as he can see. Even if it is night (Rema - or if he is blind) and he cannot see, he must distance so far that one could not see it during the day. If it is to the side, it is as if it is in back of him.

Taz (1): This is not only for Kri'as Shema. It is also for Divrei Torah. This is even if there is a Safek if there is excrement there, i.e. in front of houses where it is commonly found, and all the more so in a Chatzer.

Mishnah Berurah (3): The Pri Megadim says that the Rashba forbids speaking Divrei Torah as far as he can see, but permits thinking Divrei Torah more than four Amos from where the smell ends. Perhaps the Rosh forbids mid'Oraisa or mid'Rabanan.

Magen Avraham (2): It seems to me that the sides in front of him are like in front of him, like we say in Siman 128 (about Birkas Kohanim).

Mishnah Brurah (5): One may not say Kidush Levanah if smelly wastewater is in his field of vision. It is better not to say Kidush Levanah at all than in such a place. One who is careful not to say Divrei Kedushah in filthy places will merit long life.

Kaf ha'Chayim (37): It is forbidden even if the excrement is covered.

Doleh u'Mashke (ha'Gaon R. Chaim Kanievsky, Shlita p. 64) forbids only if he can smell or see the garbage.

Halichos Shlomo (ha'Gaon R. S. Z. Auerbach 20:5): One may Daven near a child with a dirty diaper if one cannot smell it.