You may assume that the standard laws for addition and multiplication of matrices are valid.

(a) Show that R is a commutative ring (you only need to check closure and commutativity of multiplication).

(b) Find all units of R, and all nilpotent elements of R.

(c) Find all idempotent elements of R.

**2.**
Let R be the ring **Z**_{2} [x] / < x^{2} + 1 >.
Show that although R has 4 elements, it is not
isomorphic
to either of the rings **Z**_{4} or **Z**_{2}
**Z**_{2}.

**3.**
Find all
ring homomorphisms
from **Z**_{120} into **Z**_{42}.

**4.**
Are **Z**_{9} and **Z**_{3}
**Z**_{3}
isomorphic
as rings?

**5.**
In the group **Z**_{180}^{×} of
units
of the ring **Z**_{180},
what is the largest possible order of an element?

**6.**
For the element a = (0,2)
of the ring R = **Z**_{12} **Z**_{8},
find Ann (a) = { r in R | ra=0 }.
Show that Ann (a) is an ideal of R.

**7.**
Let R be the ring **Z**_{2} [x] / < x^{4} + 1 >,
and let I be the set of all congruence classes in R
of the form [ f(x) (x^{2} + 1)].

(a) Show that I is an ideal of R.

(b) Show that R/I **Z**_{2} [x] / < x^{2} + 1 >.

*Hint*: If you use the fundamental homomorphism theorem, you can do the
first two parts together.

(c) Is I a prime ideal of R?

**8.**
Find all maximal ideals,
and all prime ideals,
of **Z**_{36} = **Z** / 36**Z**.

**9.**
Give an example to show that the set of all zero divisors
of a ring need not be an
ideal
of the ring.

**10.**
Let I be the subset of **Z**[x] consisting of all polynomials
with even coefficients. Prove that I is a
prime ideal;
prove that I is not
maximal.

**11.**
Let R be any commutative ring with identity 1.

(a) Show that if e is an idempotent element of R, then 1-e is also idempotent.

(b) Show that if e is idempotent, then R Re R(1-e).

**12.**
Let R be the ring **Z**_{2} [x] / < x^{3} + 1 >.

(a) Find all ideals of R.

(b) Find the units of R.

(c) Find the idempotent elements of R.

**13.**
Let S be the ring **Z**_{2} [x] / < x^{3} + x >.

(a) Find all ideals of S.

(b) Find the units of R.

(c) Find the idempotent elements of R.

**14.**
Show that the rings R and S in Problem 12
and Problem 13 are isomorphic
as abelian groups, but not as rings.

**15.**
Let **Z**[i] be the subring of the field of complex numbers given by

**Z**[i] = { m+ni in **C** | m,n in **Z** } .

(a) Define :
**Z**[i] -> **Z**_{2} by
(m+ni) = [m+n]_{2}.
Prove that is a ring
homomorphism.
Find ker ()
and show that it is a principal ideal of **Z**[i].

(b) For any prime number p, define
: **Z**[i] -> **Z**_{p} [x] / < x^{2}+1 >
by (m+ni) = [ m+nx ] .
Prove that
is an onto ring homomorphism.

**16.**
Let I and J be ideals in the commutative ring R,
and define the function : R -> R/I R/J by
(r) = (r+I,r+J), for all r in R.

(a) Show that is a ring homomorphism, with ker () = I J.

(b) Show that if I + J = R, then is onto, and thus R/(I J) R/I R/J.

**17.**
Considering **Z**[x] to be a
subring
of **Q**[x],
show that these two integral domains have the same
quotient field.

**18.**
Let p be an odd prime number that is not congruent to 1 modulo 4.
Prove that the ring **Z**_{p}[x] / < x^{2} + 1 > is a
field.

*Hint:* Show that a root of x^{2} = -1
leads to an element of order 4 in the multiplicative group
**Z**_{p}^{×}.