MATH 120B: HOMEWORK 1

(DUE JAN 11 TUESDAY ON GRADESCOPE)

1. Problems from textbook

Section 18 (p.174 – 177): 5, 12, 16, 19, 20, 22, 27, 31, 41

2. Problems NOT from textbook

Problem 1. Let f : A Ñ B be a map between two sets A and B.

(a) Show that f is bijective if and only if there is a map g : B Ñ A such that f � g � idB and g � f � idA,

where idA : A Ñ A is the identity map of A (i.e., idApaq � a for all a P A) and similarly for idB.

(b) Show that if f is bijective, then the map g : B Ñ A in Part (a) is unique. (We call such a unique g the

inverse of f and f�1

def

� g.)

Problem 2. Let R and R1 be unital rings (i.e., rings that have multiplicative identities). A map ϕ : R Ñ R1

is said to be a unital ring map if

(RH1) ϕpa � bq � ϕpaq � ϕpbq for all a,b P R;

(RH2) ϕpabq � ϕpaqϕpbq for all a,b P R;

(URH) ϕp1Rq � 1R1.

We also say that a unital ring map ϕ : R Ñ R1 is an isomorphism of unital rings if there is a unital ring

map ψ : R1 Ñ R such that ϕ � ψ � idR1 and ψ � ϕ � idR.

Show that a unital ring map ϕ : R Ñ R1 is an isomorphism of unital rings if and only if it is bijective.

1

- 1. Problems from textbook
- 2. Problems NOT from textbook