As mentioned in the paragraph above it, the chapter is a review, i.e. it's trying to quickly refresh the reader's memory on things they're already supposed to know; it's not trying to teach something new. So the sentence (“The set R of real numbers has two operations +: R × R → R (addition) and ∗: R × R → R (multiplication) satisfying properties that make R into an abelian group under +, and R − {0} = R∗ into an abelian group under ∗”) seems fine for that purpose.
If you look at any of the later chapters that are trying to teach something new, they are much more gentle and motivate the topic of that chapter: see e.g. “24.1 Affine Spaces” on page 759, or “26.1 Why Projective Spaces?” on page 823, etc.
Other chapters that are meant as a review are similarly terse and quick to the point (like Chapter 2), e.g. Chapter 37 “Topology” on page 1287.
I think it's good when books make conscious choices about what they're teaching versus assuming as a prerequisite (and communicate it to the reader, by using terms like “reviewed” — presumably the yet-to-be-written Introduction chapter will also mention this more explicitly).
If you look at any of the later chapters that are trying to teach something new, they are much more gentle and motivate the topic of that chapter: see e.g. “24.1 Affine Spaces” on page 759, or “26.1 Why Projective Spaces?” on page 823, etc.
Other chapters that are meant as a review are similarly terse and quick to the point (like Chapter 2), e.g. Chapter 37 “Topology” on page 1287.
I think it's good when books make conscious choices about what they're teaching versus assuming as a prerequisite (and communicate it to the reader, by using terms like “reviewed” — presumably the yet-to-be-written Introduction chapter will also mention this more explicitly).