Advanced Abstract Algebra - Maharshi Dayanand University, Rohtak

80
ADVANCED ABSTRACT ALGEBRA
xy = 0 ⇒ xyy− 1 = 0. y− 1
⇒ x.e = 0
⇒ x = 0
Hence xy = 0 ⇒ x = 0 or y = 0 and so F is without zero divisor.
Remark. It follows from this theorem that every field is an integral domain. But the converse is not true.
For example, ring of integers is an integral domain but it is not a field.
Theorem:
Any finite integral domain is a field.
Proof:
Let D be a finite integral domain
let D * = D - (o).
Since cancellation law holds in integral domain D. Since D is finite set, so one-to-one function
set to itself must be onto, so f is onto. Hence
from finite
*
∃ a ∈ D such that f a =
*
i. e. da = 1,
a ∈D CD
bg 1.
le Df
Kf
ψ
and so d is invertible. Hence every non-zero element in D is invertible, i.e. D is a field.
Remark:
Does there exist an integral domain of 6 elements No, we shall explain in Unit V that every finite integral
domain must be p n , for some prime p, every + ve integar n.
Ring homomorphism:
Let R and S be rings. A function
ψ : R
⎯⎯⎯
→ S
is called ring homomosphism
b g bg bgand
if ψ a + b = ψ a + ψ b
for all
a, b∈R.
Kernel of ring homomorphism:
is called the zero elements of S.
, the kernel of
, denoted by ker
.