Serving G-d with One's Heart and a Woman's Obligation to Pray -Berachot 20b

Today's daf states that women are obligated in תפילה because דרחמי נינהו, Tefilla is asking Hashem for mercy which women would need to do just like men.The question is what is the halakhic status of this Tefilla asking G-d for mercy? Is this considered to be Torah ordained or merely rabbinic in nature?

The Rambam in ספר המצוות lists as a Torah commandment of עבודה שבלב, to serve G-d with one's heart. He then continues by asking what is this עבודה שבלב? Tefilla.

The Rambam reiterates this idea in Mishneh Torah where he writes:

רמב"ם הלכות תפילה ונשיאת כפים פרק א

הלכה א 

מצות עשה להתפלל בכל יום שנאמר ועבדתם את ה' אלהיכם, מפי השמועה למדו שעבודה זו היא תפלה שנאמר ולעבדו בכל לבבכם אמרו חכמים אי זו היא עבודה שבלב זו תפלה, ואין מנין התפלות מן התורה, ואין משנה התפלה הזאת מן התורה, 

It is a positive Torah commandment to pray every day, as [Exodus 23:25] states: "You shall serve God, your Lord." Tradition teaches us that this service is prayer, as [Deuteronomy 11:13] states: "And serve Him with all your heart" and our Sages said: Which is the service of the heart? This is prayer.
The number of prayers is not prescribed in the Torah, nor does it prescribe a specific formula for prayer. Also, according to Torah law, there are no fixed times for prayers. (Translation courtesy of Chabad.org Mishneh Torah (English)) 

The Ramban argues on this. He says that daily תפילה is not מדאורייתא at all. Rather it is the kindness of Hashem that he accepts our prayers. The only מצוה is to daven in an עת צרה, a time of distress. In times of trouble, we as a community should turn our hearts towards Hashem in prayers.

The following excerpt on Prayer in the Teachings of Rav Soloveitchik based on a lecture by Harav Aharon Lichtenstein sheds light on this argument.

The Rav's daring comment on this debate ran as follows: the Rambam fundamentally agrees with the Ramban. Indeed, tefilla is obligatory only "in times of trouble," but the Rambam perceives man as existing in a perpetual state of crisis. Were it not for God, he could not exist for a single moment, and there can be no greater trouble imaginable than a person who is, heaven forfend, disconnected from God. Hence, we may deduce that the individual is in a constant state of crisis and needs God's contact and His mercy every day.

This argument between the Rambam and Ramban actually has a practical application a נפקא מינה להלכה in regards to a woman's obligation to pray.

According to Rambam, women are obligated in the Torah obligation of prayer which is to pray once a day. However, women are exempt from the later rabbinic enactments about the time of prayer, number of prayers, and the form and structure that prayer takes since this is all מצוות עשה שהזמן גרמה. However, according to the Ramban the entire obligation of daily prayer is rabbinic. The rabbis who obligated us to pray 3 times a day included the obligation to say שחרית ומנחה for women as well since it is בקשת רחמים.

The Mishna Berurah ( קו:ד) follows the opinion of the Ramban and writes that women should daven שמונה עשרה of שחרית ומנחה daily as well as the first פסוק of שמע, אמת ויציב until שמונה עשרה in order to be סומך גאולה לתפילה. But women do not have to say מעריב because it is a רשות, optional. Women are also obligated to say ברכות השחר ברכות התורה and פסוקי דזמרה. A woman can fulfill her ברכת התורה by saying אהבה רבה.

A woman who is so involved in child rearing that she has no time to daven should recite a short תפילה that includes שבח בקשה והודאה. The son of the חפץ חיים writes about his mother that she hardly ever davened שמונה עשרה when they were little and the חפץ חיים exempted her becasue she was so involved in raising the children. This short תפילה for women could be ברכות השחר and ברכות התורה which both contain these elements. The 3 blessings of ברכות התורה include these:

Praise-שבח

  ברוך אתה ה' אלהינו מלך העולם אשר קדשנו במצותיו וצונו לעסוק בדברי תורה

Blessed are you Hashem, our G-d, King of the Universe, who has  made us holy through his commandments, and has commanded us to engage in the study of the words of Torah. 

Request-בקשה

  הערב נא ה' אלהינו את דברי תורתך בפינו ובפה עמך בית ישראל ובצאצאינו ובצאצאי עמך בית ישראל 

ונהיה כלנו וצאצאינו מלומדי תורתך ויודעי שמך ברוך אתה ה' המלמד תורה לעמו ישראל Please Hashem our G-d, make the words of Your Torah sweet in our mouths and in the mouths of Your people, the house of Israel, so that we, our descendants, and the descendants of Your people, the house of Israel, may all know and study your Torah for its own sake. Blessed are you Hashem who teaches Torah to his nation Israel. 

Thanks-הודאה

 ברוך אתה ה' אלהינו מלך העולם אשר בחר בנו מכל העמים ונתן לנו את תורתו. ברוך אתה ה' נותן 

התורה

Blessed are You, Hashem our G-d, King of the Universe, who has chosen us from all the peoples and given us His Torah. Blessed are you, Hashem, Giver of the Torah. (Translation from the Koren Sacks Siddur.)