Math 653
Homework #4
November 6, 2012
Due Thursday, November 15.
Justify all of your work.
Notation. Let R be a ring. Suppose S ⊆ R is a subring and α ∈ R. Then
S[α] = {r0 + r1 α + · · · + rn αn | ri ∈ R, n ≥ 0}.
It is easy to check that S[α] is subring of R, and not surprisingly it’s called the subring generated
by S and α. Also, for α1 , . . . , αm ∈ R, we can similarly define the subring S[α1 , . . . , αm ] ⊆ R by
taking polynomials in α1 , . . . , αm .
Problem E1: Let R be a commutative ring with 1. Let R[x] be the polynomial ring in the variable
x over R, and let R[x, x−1 ] be the ring of Laurent polynomials in x over R:
( n
)
X
−1
i R[x, x ] =
ai x ai ∈ R, n ∈ N .
i=−n
(a) Let G be a finite cyclic group of order n. Show that R[G] ∼
= R[x]/(xn − 1) as rings.
(b) Let G be an infinite cyclic group (G is isomorphic to the additive group of Z). Show that
R[G] ∼
= R[x, x−1 ].
Problem E2: Let F be a field, and let R = F [x2 , x3 ], which is a subring of the polynomial ring
F [x]. Show directly that R is not a principal ideal domain.
Problem E3: The center of a ring R is the subring Z(R) = {z ∈ R | zr = rz, ∀r ∈ R}.
(a) What is the center of Mat2 (Q), where Q is a field?
(b) What is the center of Q[S3 ], where S3 is the symmetric group on {1, 2, 3}?
Problem E4: Let H be a 4-dimensional real vector space with basis labeled {1, i, j, k}. We make
H into a ring as follows. Define multiplication of basis elements by
i2 = j 2 = k 2 = −1;
ij = −ji = k;
jk = −kj = i;
ki = −ik = j.
We specify that any a ∈ R commutes with each basis element, and then define multiplication of
general elements of H by extending by linearity. You should convince yourself that H is a ring.
(a) Show that H is a division ring. The ring H is called the ring of real quaternions.
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(b) Let R be the set of all 2 × 2 matrices of the form
subring of Mat2 (C) and that R ∼
= H.
α β
−β α
with α, β ∈ C. Show that R is a
(c) Let K be a 4-dimensional complex vector space with basis {1, i, j, k}. Using the same procedure as in the definition of H, define a ring structure on K. Show that K is not a division
ring. Moreover, show that K is isomorphic as a ring to Mat2 (C).
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