algebra vs pure mathematics

algebra and pure mathematics both are nouns.

Word	Noun	Adjective	Verb	Adverb
algebra	Yes	No	No	No
pure mathematics	Yes	No	No	No

As nouns, pure mathematics is a hypernym of algebra; that is, pure mathematics is a word with a broader meaning than algebra:

algebra: the mathematics of generalized arithmetical operations
pure mathematics: the branches of mathematics that study and develop the principles of mathematics for their own sake rather than for their immediate usefulness

algebra (noun)	pure mathematics (noun)
the mathematics of generalized arithmetical operations	the branches of mathematics that study and develop the principles of mathematics for their own sake rather than for their immediate usefulness