Traditional Jewish views
According to traditional Jewish law, mere belief in the principles of Judaism does not make one a Jew. Similarly, non-adherence by one who is Jewish to Jewish principles of faith does not make one lose one's Jewish status.
Jewishness is determined by the mother (matrilineal descent); thus the immediate male descendants of a female Jewish apostate are still considered Jewish; all her female descendants, but only in a documented unbroken female line of descent, and their immediate male children are also considered Jewish. While most of these descendants probably would not be practicing Judaism, or in many cases aware of their Jewishness, their status as Jews technically still would be in effect. As such, all Jewish denominations welcome the return of any of these people back to the Jewish community; such people would be considered Jews in good standing without the need for a formal conversion.
Generally, people who have been raised as non-Jews (called Gentiles) would be expected to make some sort of public sign that they are returning to Judaism, for instance engaging in a course in Jewish Torah education, joining a synagogue, observing the Jewish Shabbat (Sabbath), the Jewish festivals, keeping kosher, commencing "family purity" or niddah, having an adult bar mitzvah ceremony, and anything else they should try to observe. If not circumcised, males are required to have a brit milah (ritual circumcision, the ceremony is called a "bris" by Ashkenazi Jews.)
Note that "circumcision" in the Jewish sense is not the medical procedure performed by a doctor but is a religious procedure performed by a mohel (also pronounced as mo'el).
Jewish peoplehood is not inherited from one's Jewish father alone, even if he were not an apostate from Judaism. This traditional rabbinic view is still held by many in the return-to-tradition wing of Reform Judaism, and by all of Orthodox Judaism and Conservative Judaism.
View of the State of Israel
The situation in Israel is somewhat ambiguous. One area where the definition of Jew is relevant is in deciding who qualifies to make aliyah under the Law of Return. The requirements here differ significantly from the definition of a Jew under halakha, as the law attempts to include all those who might be subject to anti-Semitic persecution, including anyone with a Jewish grandparent, as well as non-Jewish spouses of Jews. However it specifically excludes Jews who have converted to a faith other than Judaism.
A second area where the definition of Jew is relevant is in marriages and divorces, which are under the jurisdiction of the Israeli Ministry of the Interior that, unlike the Law of Return, defines Jews strictly according to halakha.
A third relevant area is in the registering of "nationality" on Israeli identity cards. This is also controlled by the Ministry of the Interior, which has generally only registered as a "Jew" those who meet the halakhic definitions. However, in a small number of cases the Supreme Court of Israel has forced the Ministry to register individuals who did not meet that definition.
See also: Jew in Israel and Israeli Law
Views of Reform and Reconstructionist Judaism
In the last half of the 20th century, two theologically liberal (primarily American) Jewish groups—Reform Judaism and Reconstructionist Judaism—have allowed people who do not meet the classical halakhic criteria to define themselves as Jews. They no longer require converts to follow traditional Jewish procedures of religious conversion, and they accept a person as a Jew even if their mother is non-Jewish; in the case of Reform, so long as the father is a Jew and the person performs "appropriate and timely public and formal acts of identification with the Jewish faith and people". This is commonly (though mistakenly) known as patrilineal descent.
This has thus resulted in a serious schism among the Jewish people; today many Reform Jewish and secular American Jews born from originally Gentile mothers, (who were not born Jewish themselves), consider themselves to be Jews, although they are not considered Jewish by Orthodox Judaism, Conservative Judaism, and even by many Reform Jews outside of the United States.
Some Reform Jews view Judaism as a religion alone, and thus they view Jews who convert to another faith as non-Jews. This contrasts to the traditional rabbinic view of Judaism as a peoplehood, and not merely a religion. In the traditional view, those who leave Judaism by converting to another religion are still seen as Jewish people; however, they are seen as apostates who by their actions have chosen to remove themselves from the Judaic religion.
Maintaining Jewishness versus assimilation
Judaism guides its adherents in both practice and belief, and has been called not only a religion, but also a "way of life," which has made the job of differentiating between Jews and Judaism almost impossible.
In many times and places, such as the Hellenization during ancient
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