PREP PROBLEMS

1 Elementary divisibility problems

Let R be an integral domain and x, y, z ∈ R.

1. (a) Show that .
(b) Show that x|y and for some u ∈ U(R).

2. Show that .

3. Show that and z is a linear combination of x
and y .

4. Show that if lcm (x, y) exists, then gcd(x, y) exists, and moreover,
gcd(x, y)lcm(x, y) = xy.

5. If gcd(x, y) exists, does lcm(x, y) necessarily exist?

6. Let z ≠ 0. Show that if gcd(xz, yz) exists, then gcd(x, y) exists, and
moreover, gcd(xz, yz) = zgcd(x, y).

7. Show that if x is prime, then x is irreducible. Give examples to show that
the converse is false.

8. Show that every two elements of R have a gcd every two elements of
R have an lcm. Such an integral domain R is called a GCD-domain.

9. Show that in a GCD-domain, every ir reducible element is prime.

10. Show that R is a UFD R is atomic (i.e., every non zero nonunit of
R is a product of irreducible elements) and a GCD-domain. Thus every
irreducible element of a UFD is prime.

2 Prime and irreducible elements

1. Show that Z + XQ[X] is not atomic.

2. Find the irreducible and prime elements of Z + XQ[X].

3. Find an infinite family of prime elements of Z + XQ[X] such that


4. Let K be proper subfield of a field L. Find the irreducible and prime
elements of K + XL[X].

5. Let K be a field. Find the irreducible and prime elements of K[X²,X³].

6. Does have any irreducible elements?

7. Does have any irreducible elements?

3 Monoid domain problems

Let R be a commutative ring and S an additive commutative monoid. Define
the monoid ring , almost all
0 } with and multiplication defined by
(and then extend using distributivity ).

For example: , and R[X; {0, 2, 3, . . .}]
= R[X²,X³].

1. Let S be a submonoid of a torsionfree abelian group G . Show that S is
cancellative and torsionfree. (A monoid S is torsionfree if nx = ny for
x, y ∈ S and n ≥1 => x = y.)

2. Let S be a torsionfree commutative cancellative monoid. Show that S is
a submonoid of a torsionfree abelian group. (Thus S is a submonoid of
some .)

3. Show that an abelian group can be totally ordered if and only it is torsionfree.

4. Let G be an abelian group. Show that R[X;G] is an integral domain
R is an integral domain and G is torsionfree.

5. Show that R[X; S] is an integral domain R is an integral domain and
S is a torsionfree commutative cancellative monoid.

6. Let R[X; S] be an integral domain. Show that 
. (Here is the group of elements of S
with inverses.)

4 Localization problems

Let R be an integral domain with quotient field K. Let be a
n onempty multiplicatively closed set. The localization of R at S is
K | r ∈ R, s ∈ S }. Define = { x ∈ R | xy ∈ S for some y ∈ R}. Clearly
. We say that S is saturated if = S.

1. Show that is a subring of K containing R.

2. Show that . Thus is multiplicatively closed and
saturated.

3. Show that .

4. Let P be a prime ideal of R and let S = R \ P. Show that = S and
that is an integral domain with a unique maximal ideal
p ∈ P, s ∈ R \ P }. This ring is denoted by .

5. Let be a set of prime elements of R, and let 
u ∈ U(R), p_i ∈ P }. We say that S is generated by prime elements of P.

(a) Show that S is a saturated multiplicatively closed set.

(b) Suppose that R is atomic. Show that is atomic and has no
prime elements.

(c) Show that satisfies ACCP (resp., is a BFD, an HFD, an FFD) if
R satisfies ACCP (resp., is a BFD, an HFD, an FFD).

6. Give an example of an integral domain R which is not atomic and a
multiplicatively closed subset S of R generated by prime elements of R
such that is a UFD (and hence is atomic, satisfies ACCP, is a BFD,
an HFD, and an FFD).

7. Let be a set of prime elements of R such that (i) for
all and (ii) for any (infinite) sequence of nonassociate
primes in P, and let . Show that
R is atomic (resp., satisfies ACCP, a BFD, an HFD, an FFD, a UFD) if
is atomic (resp., satisfies ACCP, a BFD, an HFD, an FFD, a UFD).

8. Let R = K[X; T], where K is a field and
is an additive submonoid of . Show that R is a BFD, and hence R
satisfies ACCP and R is atomic. Is R an HFD or an FFD? Let S = {X^q |
0 ≠ q ∈ T }. Show that RS is not atomic, and hence R does not satisfy
ACCP and is not a BFD.