"The woman has milk."
Translation:У женщины есть молоко.
женщина -> nominative (used for the subject in normal sentences) женщины -> genitive (coincides with the plural nominative like most feminine nouns), used whenever genitive is used (after "у", after "для", to express non-existance with "нет", to show possesion like english " 's ", and many other places, most of the time when English could use "of")
Here the genitive is used because of "у":
"У <X in the genitive> есть <Y in nominative>" means "X has Y". Literally, the sentence says something like "By the side OF X, there is Y", so Y works as a subject of the verb to be and so goes in nominative. But this "By the side of X there is Y" is a Russian idiom that expresses the English "to have" construction.
The suggested translation was "у зтой женщины есть молоко" but it is not accepted. Is it truly incorrect?
Well, it's hard to say, often the article "the" translated as "эта, этот, эти, это, этой, этого, этих, and so on". Literally means "That woman has milk".
For me it seems that the origins of the definite articles are the "this" and "that" words.
I guess that "у женщины есть молоко" can be translated as "a woman/the woman" while "у этой женщины есть молоко" as "the woman/this woman/that woman"
I don't think the English lessons are advising you correctly then. Эта женщина usually means "this woman", rather than "the woman". In English, we use "this" or "that" to distinguish one particular woman out of a group. We use "the" if there is only one woman in the picture. Without more context to this sentence, it seems there is only one woman. Ясно?