Description
Let R be a ring with identity 1 6= 0.
7.3.1 Prove that the rings 2Z and 3Z are not isomorphic
7.3.10 Decide which of the following are ideals of the ring Z[x]:
- the set of all polynomials whose constant term is a multiple of 3.
- the set of all polynomials whose coefficient of x2 is a multiple of 3.
- the set of all polynomials whose constant term, coefficient of x and coefficient of x2 are zero.
- Z[x] (i.e., the polynomials in which only even powers of x appear).
- the set of polynomials whose coefficients sum to zero.
- the set of polynomials p(x) such that p0(0) = 0, where p0(x) is the usual first derivative of p(x) with respect to x.
2
7.3.26 The characteristic of a ring R is the smallest positive integer n such that
1 + 1 + … + 1 = 0
| n times{z }
in R; if no such integer exists the characteristic of R is said to be 0. For example, Z/nZ is a ring of characteristic n for each positive integer n and Z is a ring of characteristic
0.
(a) Prove that the map Z→R defined by
|
1 + 1 + … + 1 (k times) k 7→ 0 |
if k > 0 if k = 0 |
−1 − 1 − … − 1 (−k times) if k < 0
is a ring homomorphism whose kernel is nZ, where n is the characteristic of R (this explains the use of the terminology “characteristic 0” instead of the archaic phrase “characteristic ∞” for rings in which no sum of l’s is zero). (b) Determine the characteristics of the rings Q, Z[x], Z/nZ[x].
(c) Prove that if p is a prime and if R is a commutative ring
of characteristic p, then
(a + b)p = ap + bp for all a,b ∈ R.
4
7.3.28 Prove that an integral domain has characteristic p, where p is either a prime or 0 (cf. Exercise 26).
7.4.10 Assume R is commutative. Prove that if P is a prime ideal of R, and P contains no zero divisors then R is an integral domain.
Problem A
- Let R be an integral domain. As you conjectured in class, prove that the units in R[x] are precisely the constant polynomials p(x) = u where u is a unit in R.
- On the other hand, show that p(x) = 1 + 2x is a unit in R[x], where R = Z/4Z.
