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 = a^p + b^p 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.