To: Ippolita
By my understanding, if a Jewish man marries a gentile woman, the children are considered gentile, but if a Jewish woman marries a gentile man, the children are accepted as Jewish.
That would be a matriarchal structure, though I've never actually asked a Jewish person if what I've said above is true.
Also, I understand that people can be Jewish just by deciding that the God of Abraham is their God (just as Christians and Moslems do), so children of Jewish men could be Jewish if they want to, but Jewish women's children would be Jewish by Jewish law, without choice in the matter.
This is shown quite clearly in the Bible. Ishmael could not be part of the Jewish story because his mother was not Jewish, while Isaac was Jewish because Sarai was. If the line had been traced throught the male, in this case Abraham, then Ishmael would have been as Jewish as Isaac.