I am self-studying the following notes: Introduction to Commutative algebra and algebraic geometry.
There is a sudden run of statements from end of page 6 / beginning of page 7 which leave me confused. I have numbered the statements for clarity:
is $(2)$ a contraction as per Definition 1.17 ? what is the domain and target of the map $(2)$ anyway ?
how does this map $(2)$ induce $(3)$ ?
Does map $(4)$ refer to the Nullstellentstaz (weak form) as stated in statement $(0)$ / theorem 1.16 ?
how would $(4)$ induce map $(5)$ ?
Many thanks for any guidance.