Shevi'is, chapter 5, mishnah 10. The mishnah opens with the law of lending utensils to someone suspected of not keeping shemittah properly: "Mash'eles ishah la'chaverta ha'chashudah al ha'shevi'is" - this refers to a neighbor who is suspected of not observing the laws of shemittah, meaning that she uses shemittah produce after the time of bi'ur without performing bi'ur on it. Even so, she is permitted to lend her the following utensils:
"Nafah u'kevarah" - two different types of sieves.
"Ve'rechayim" - a millstone for grinding.
"Ve'sanur" - an oven for baking.
The permission stems from the fact that we can assume she is using the utensils for permitted purposes and not for forbidden ones, and we are entitled to rely on that assumption.
The limit - the prohibition of assisting those who transgress:
However, she may not become actively involved together with her, because that constitutes assisting those who transgress - direct help to people who are doing something forbidden or are likely to do so. Therefore the mishnah says: "Aval lo tavor ve'lo tit'chan imah" - she may not sort together with her, that is, separate the gravel and stones from the grain, and she may also not grind the grain together with her, since it is forbidden to give direct assistance to transgressors.
The wife of a chaver and the wife of an am ha'aretz - the laws of tevel:
From here the mishnah moves from the subject of shemittah to the laws of tevel, the separation of Terumah and tithes: "Eishes chaver mash'eles le'eishes am ha'aretz" - the wife of a chaver, a Torah scholar who is scrupulous about the laws of Terumah and tithes and about the laws of tumah and taharah, is permitted to lend to the wife of an am ha'aretz. And not only is she permitted to lend her a "nafah u'kevarah", the two types of sieves mentioned above, but "u'voreres ve'tochenes u'merakedes imah" - she may even be directly involved together with her in sorting the kernels and separating them from the chaff and the stones, in grinding with her, and even in sifting the flour with her.
The reason for this: even though amei ha'aretz are suspected of not separating tithes, nevertheless "rov amei ha'aretz me'asrin". The principle that most amei ha'aretz do in fact separate tithes remains in force, and therefore she is permitted to work together with her directly.
"Aval mishetatil es hamayim - lo tiga etzlah": - But once she has poured in the water, she may not touch anything of hers.
From the moment she adds water to the flour, the wife of the chaver may not touch anything belonging to her. Since the wife of the am ha'aretz does not keep the laws of tumah and taharah, her utensils are impure and not pure. Therefore, if the wife of the chaver were to handle the dough directly, the dough would become impure from the utensils of the wife of the am ha'aretz, it would be impossible to separate challah in purity, and other problems would arise as well. She cannot assist the activities of those who act improperly, and here they are causing tumah to the challah, and therefore she may not be directly involved in it.
The reason for these leniencies - for the sake of peace:
All these leniencies, both the lending of utensils and working together with her to a certain degree, were permitted only "mipnei she'yihyeh shalom bein Yisrael" - so that there should be peace among Israel. Were it not for the matter of peace, we would not want to have any connection to this at all, but because they sought to promote peace they permitted certain things.
This principle was already taught earlier in the tractate: we may encourage non-Jews during the shemittah year, that is, we may offer them a blessing or a "yishar ko'ach", even though it is the shemittah year, since they are not commanded in the laws of shemittah. But a Jew who does not keep the laws of shemittah and works his land during the seventh year - him we certainly do not encourage, and we do not offer him a "yishar ko'ach".
And returning to non-Jews: it is permitted to greet them and to inquire how they are feeling, and even to wish them peace. And as Chazal explained, one may do so even on their festival day - and again, for the sake of peace.
In summary: in this mishnah we learned the line between lending utensils to one suspected regarding shemittah, which is permitted because we attribute the use to a permitted purpose, and direct assistance in sorting and grinding, which is forbidden as assisting those who transgress. We also learned the law of the wife of a chaver and the wife of an am ha'aretz regarding tevel, where even direct involvement is permitted because "most amei ha'aretz do separate tithes", and the limit set once the water has been added - because of the tumah of the utensils and the harm to separating challah in purity. And finally we saw that all these leniencies, as well as encouraging non-Jews during shemittah and greeting them even on their festival day, were stated only for the sake of peace.