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Math 4370 – Homework 1
Read: Read Chapter 1 from the textbook as well as the course notes.
Turn in:
1. Let R be a ring with two operations + and ×. Let 0 and 1 denote the identity elements for + and ×. If 1 = 0 show1 that R = {0}.
2. Let n be any integer. Consider the set Z/nZ consisting of integers modulo n with the addition and multiplication modulo n.
(a) Show Z/nZ forms a ring with n elements.
(b) Let p be a prime (integer) number. Show Z/pZ is indeed a field. We often denote this field by Fp.
(c) Prove that a p = a for all a ∈ Fp.
(d) Find a nonzero polynomial in Fp[x] which vanishes at every point of Fp.
3. Let R be an integral domain. Prove:
(a) deg(p(x)q(x)) = deg(p(x)) + deg(q(x)) for all p(x), q(x) ∈ R[x].
(b) R[x] × = R×, i.e., R and R[x] have the same units.
(c) R[x] is an integral domain.
If R contains zero divisors, could any of (a), (b), (c) still be true?
4. Recall a commutative ring R with 1 (identity) is called Noetherian if every ideal of R is finitely generated. Prove the following are equivalent:
(1) R is a Noetherian ring.
(2) R has no infinite increasing chains of ideals, i.e., whenever I1 ⊆ I2 ⊆ I3 ⊆ · · · is an increasing chain of ideals of R, then there is a positive integer m such that for all k ≥ m we have Ik = Im. (this is called ascending chain condition on ideals).
5. An integral domain R is said to be a Euclidean domain (or possess a division algorithm) if there is a function N : R → Z≥0 with N(0) = 0, such that for any two element a ∈ R and 0 6= b ∈ R there exist elements q and r in R with a = qb + r with r = 0 or N(r) < N(b) .
(a) Show that the ring of integers Z and the polynomial ring in one variable k[x] over a field k are examples of Euclidean Domains.
(b) Prove that every ideal in a Euclidean Domain is principal. More precisely, show that if I is a nonzero ideal in the Euclidean Domain R, then I = hdi, where d is any nonzero element of I of minimum N(·)-value.
6. Let R be a commutative ring with identity, and I ⊆ R be an ideal. Prove
(a) I = R if and only if I contains a unit of R.
(b) R is a field if and only if {0} and R are the only ideals of R.
7. Let R be a commutative ring and let I 6= R be an ideal. Recall I is called a prime ideal if whenever ab ∈ I for some a, b ∈ R then a ∈ I or b ∈ I. Also recall I is called a maximal ideal if the only ideals containing I are I and R. Prove:
(a) I is a prime ideal in R if and only if the quotient ring R/I is an integral domain.
(b) I is a maximal ideal in R if and only if the quotient ring R/I is a field.
(c) Every maximal ideal is a prime ideal.
8. Let I be an ideal of a commutative ring R and define the radical of I rad(I) = {r ∈ R: r n ∈ I for some n ∈ Z>0} .
(a) Prove that rad(I) is an ideal of R containing I.
(b) If rad(I) = I we call the ideal I radical. Show that prime ideals are radical.
(c) Prove that for any ideal I we have rad(rad(I)) = rad(I), i.e., rad(I) is radical.
Note: rad(I) is sometimes denoted by √ I. It is a theorem that rad(I) = T P where the intersection is over all prime ideals P containing I.
Notes:
(i) Collaboration: On the homework sets, collaboration is both allowed and encouraged. However, you must write up yourself and understand your own homework solutions. You should give credit to any outside sources or collaborations.
(ii) Academic honesty: All students are expected to comply with the Code of Academic Integrity. The institute honor code is available at: www.theuniversityfaculty.cornell.edu/AcadInteg/code.html
(iii) Please turn in your homework in class. If there are multiple pages, please staple them. Emailed homework will not be accepted unless you have a special reason, and you have discussed your situation with me a priori. In that case, please send a single .pdf file, put your name on all pages, and make sure to number the pages. 3