In this task you will be given a text passage from a trivia quiz. You should choose the category this question belongs to from these categories: History, Science, Social Science, Fine Arts, Literature. You can only choose one category, so if in doubt, please choose the closest match.

The Hilbert Basis Theorem guarantees that if one of these is Noetherian, then the polynomials over it are also Noetherian. While the existence of a multiplicative unit in one of these isnt guaranteed by the axioms, most authors assume its presence. These can form quotients with special subsets known as ideals, and commutative ones with no zero divisors are known as integral domains. Formally an abelian group over addition and a monoid over multiplication, this describes, for ten points, what algebraic structure exemplified by the integers
Science

Her third husband was a farmer who lost his crops to a horde of fairy mice, but they stopped harassing him after he caught one and threatened to hang it. Before she married Manawydan, she had a child by her second husband who vanished from their home and ended up at the court of Teirnon. The daughter of Hefeydd the Old, this lady of the Singing Birds decided that she preferred a lord of Dyfed to her first husband, Gwawl. Considered a version of the horse goddess Epona, she was the mother of Pryderi by the man she loved, Pwyll. FTP, name this figure from Welsh myth who rings like a bell through the night in a song by Fleetwood Mac.
Social Science

The Jocic reaction produces these compounds from trichloromethylcarbinols. The Schmidt reaction produces primary amines from these compounds and hydrogen azide, and the Jones Reaction makes them from primary alcohols. The Fischer esterification creates esters from alcohols and these compounds in acidic solution, and their namesake functional group forms the C terminal end of proteins. For 10 points, identify these compounds which have a COOH functional group.

Science