The name of this week's Torah reading is
Chayei Sarah, "the life of Sarah." This raises an obvious question. The Torah reading talks of Sarah's death and her burial, why should its name be associated with "her life"?
With this name, the Torah is teaching us that every person can gain an aspect of immortality. We are not speaking of the afterlife in the spiritual realms where every soul is granted an eternal existence, but rather a continuous posterity in this material world.
To explain: The Torah reading focuses on three events: Abraham's purchase of a burial place for Sarah in Hebron, his first acquisition of a portion of the Land of Israel, the mission to find a wife for Isaac, and Abraham's granting Isaac his inheritance.
All of these events reflect Sarah's lifework. Firstly, as a woman she endeavored that the pledge G-d gave to Abraham, that the land of Israel become the heritage of the Jewish people, not remain merely an abstract promise, but be translated into actual fact. This transpired with the purchase of the Cave of Machpelah. From that point on, the Jews owned a portion of the Holy Land, and that ownership was recognized by all the nations of the world.
She desired that her son marry and perpetuate her family. This is reflected in the choice of Rebecca as a wife for Isaac. Indeed, our Sages explain that it was when Isaac saw that Rebecca possessed the spiritual virtues of his mother that he loved her.
And when Abraham distributed his inheritance, he gave "everything he had to Isaac," giving his other children mere "presents." This reflects the influence of Sarah who told Abraham that "the son of the handmaiden will not inherit with my son, with Isaac."
And thus the Torah reading tells us of Sarah's life, for the events that it describes are a reflection of her ongoing influence, how the manner in which she affected her family and her environment was perpetuated beyond her immediate physical presence.