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 :
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 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 C-vector 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 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 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 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