Can a women write a Mezuzah?
The gemara in Gittin 45b states:
ספר תורה תפילין ומזוזות שכתבן אשה וכו' פסולין שנאמר וקשרתם וכתבתם כל שישנו בקשירה ישנו בכתיבה
A Torah scroll, phylacteries, or mezuzot [...] that was written by a woman is unfit for use, as it is stated: "“And you shall bind them… and you shall write them" anyone who's included in the mitzva of binding can write them
This would seem to indicate that a women may not write a Mezuzah. The Drisha(יור''ד סימן רפ''א) asks why is it that the Tur skips this Halacha about Mezuzah but does say it about Teffilin (אור''ח סימן ל''ט) The Rif and the Rosh also fail to mention this Halacha of women not being Kosher to write a Mezuzah.That is since the above mentioned exception to women states that only those who can not wear Tefillin cant not write it seeing that it does not apply to Mezzuzah nor Tefillin it seems that women may write them. Therefore the Drisha comes out That A WOMEN MAY WRITE A TORAH AND MEZUZAH.
The Beis Yosef the Mechaber(יור''ד סימן רפ''א) changes and says a Torah she may not but makes no mention of not allowing Mezzuzah.
In SH"UT Shagas Aryeh he says the reason that women may not write Mezzuzos is because since we learn out of the word וכתבתם the laws of writing Teffilin so to Mezzuzah from which we learn out from the Tefillin for sure a women can not write.
We are left with a Question as the Meor Ktziah (אור''ח סימן ל''ט) points out The Mechaber never says a women can not write Tefillin!!!
The Mishneh Berurah(אור''ח סימן ל''ט ס''ק א') brings that according to Halacha that a women may not write Tefillin and answers that the Mechaber did not have to write it explicitly by Mezuzah because by Teffilin he writes just as a women or slave writes Teffilin is Posul so to by Sefer Torah and Mezuzah and he did not need to repeat himself.