Mathematics G4261x Commutative Algebra Assignment #1 Due Sept. 10, 2014 Throughout the course, all rings are assumed to be commutative with identity 1, and all homomorphisms take 1 to 1. 1. Let fRi j i 2 Ig be a family of rings. Define a product on the direct product of abelian Q groups i2I Ri and show that it is a ring. What is the identity? What are the units? If the Ri are integral domains, is the same true of the direct product? 2. Define the characteristic of a ring R to be the least positive integer n such that 0 = 1 + ··· + 1 (n times), or 0 if there is no such n. (a) Show that if R is an integral domain, then the characteristic is either 0 or prime. (b) Show that a ring of characteristic 0 always has a subring ∼= Z. Show that a ring of characteristic n > 0 always has a subring ∼= Z=(n). (c) Show that a field of characteristic 0 always has a subfield ∼= Q. Show that a field of characteristic p > 0 always has a subfield ∼= Z=(p). 3. Show that in a ring R of prime characteristic p, the Frobenius map f : R ! R given by f(x) = xp is a ring homomorphism. You may use uniqueness of prime factorization for integers. 4. Show that Z is a principal ideal domain. 5. Determine, with proof, all of the irreducible polynomials in R[x] up to units. You may use the fundamental theorem of algebra. 6. An element of a ring is nilpotent if an = 0 for some n > 0. Prove that the nilpotent elements of a ring R form an ideal N. Prove that the quotient ring R=N has no nonzero nilpotents. Show that R has no nonzero nilpotents if and only if 0 is the only element whose square is 0. 7. For an integral domain R, define the field of fractions to be the set of equivalence classes of R × (Rnf0g), where (a; b) is denoted a=b, and the equivalence is (a; b) ∼ (c; d) if ad = bc. Show, if addition and multiplication are defined as usual for fractions, that this is indeed a field. Show that the field of fractions of Z is Q, and that that of a field K is K itself. What goes wrong if R is not a domain, such as Z=(6)? All documents pertaining to the course, including this one, will soon be available from http://www.math.columbia.edu/~thaddeus/commalg.html.