H OMEWORK 3
M ATH 120B (45040) W INTER 2012
Due Thursday, February 2nd at the beginning of discussion section.
1. a. Let R be a eld and let : R S be a ring homomorphism. Show that is either the zero
map or is injective.
b. Find an example of a ring R, a e
H OMEWORK 9
M ATH 120B (45040) W INTER 2012
Due Thursday, March 15th.
1 0
. (Remember, over a noncom1. Find all ideals of M22 (R) which contain the matrix
0 0
mutative ring, ideals must be closed under multiplication on both sides by elements in the ring.
H OMEWORK 2
M ATH 120B (45040) W INTER 2012
Due Thursday, January 26th at the beginning of discussion section.
1. Let R denote a ring. An element r R is called nilpotent if there exists a positive integer n
such that rn = 0.
a. Show that a nonzero nilpote
H OMEWORK 8
M ATH 120B (45040) W INTER 2012
Due Thursday, March 8th.
1. Show that C[x]/(x2 ) and C[x]/(x) C[x]/(x) are isomorphic as Cvector spaces but not isomorphic as rings.
2. (This was on the rst draft of our midterm.) Let R denote a ring and let S
H OMEWORK 4
M ATH 120B (45040) W INTER 2012
Due Thursday, February 9th.
1. Let F be a eld and let be the function from F [x] to F which sends a polynomial an xn +
an1 xn1 + + a0 to the sum of its coefcients. Is a ring homomorphism? What about the
function
H OMEWORK 7
M ATH 120B (45040) W INTER 2012
Due Thursday, March 1st.
1. Let S R denote a subset that is closed under addition and is closed under multiplication by
arbitrary elements in R. Must S be an ideal? What if we further require that 1 R.
2. Let :
.
Student ID:
Name:
MATH 120B, Winter 2013
Midterm, Feb 8, 2013
Show all work. If you detach any sheets from the exam, yoUmust put your name and ID number
on the detached sheets.
Question 1
Question 2
Question 3
Question 4
Question 5

Total
1
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1. (1)10