Abstract Algebra (Part 2)

Problem 1

Let I be an ideal in the integers which contains 18 and 30.  What are the possibilities for I?

Problem 2

How many nilpotent elements does (Subscript[ℤ, 72]) have?

Problem 3

If today is Tuesday, what day of the week will it be in (10^1100) days?

Problem 4

In the ring (Subscript[ℤ, 5])[x], what do the elements of the ideal <(x^2)+6> and the factor ring (Subscript[ℤ, 5])[x]/<(x^2)+6> look like? (hint: polynomial long division works over any field)  Determine if the ideal <(x^2)+6> is maximal, prime, or neither.

Problem 5

Explain why the set {(x,x)|x∈ℤ} is a subring of ℤ⊕ℤ but not an ideal of ℤ⊕ℤ.

Problem 6

Define α:ℝ[x]->ℂ by α(f(x))=f(i).  Show that α is a ring homomorphism, find its range and kernel, and give the corresponding statement about a factor ring.  What does this statement tell you about the kernel?

Problem 7

If α is a ring homomorphism from ℤ⊕ℤ to itself, what are the possible values of α((0,1))?

Problem 8

We commonly use the same divisibility language for commutative rings as we do for integers.  We say a|b in a commutative ring R if ∃r∈R, b=a r.  Show that in an integral domain D, for a,b∈D   a|b and b|a  if and only if a and b are associates.

Problem 9

An element p in an integral domain D is called prime if p|(a b)⟹ p|a or p|b.  Explain why 5 is prime in ℤ with this definition but not prime in ℤ[i] (and therefore “being prime” is relative to the set you are talking about).

Note:  What you are used to thinking of as “prime” is in ring theory called irreducible.  An element x of integral domain is irreducible if x= a b implies either a is a unit or b is a unit.  Prime elements defined above are always irreducible, but there are example of integral domains in which there are irreducibles which are not prime.  In a principal ideal domains (which includes ℤ where arithmetic usually takes place) the idea of primes and irreducibles are the same.