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2011 | OriginalPaper | Chapter
Normed Rings
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Norms ‖⋅‖ of associative rings are generalizations of absolute values |⋅| of integral domains, where the inequality ‖
xy
‖≤‖
x
‖⋅‖
y
‖ replaces the standard multiplication rule |
xy
|=|
x
|⋅|
y
|. Starting from a complete normed commutative ring
A
, we study the ring
A
{
x
} of all formal power series with coefficients in
A
converging to zero. This is again a complete normed ring (Lemma 2.2.1). We prove an analog of the Weierstrass division theorem (Lemma 2.2.4) and the Weierstrass preparation theorem for
A
{
x
} (Corollary 2.2.5). If
A
is a field
K
and the norm is an absolute value, then
K
{
x
} is a principal ideal domain, hence a factorial ring (Proposition 2.3.1). Moreover, Quot(
K
{
x
}) is a Hilbertian field (Theorem 2.3.3). It follows that Quot(
K
{
x
}) is not a Henselian field (Corollary 2.3.4). In particular, Quot(
K
{
x
}) is not separably closed in
K
((
x
)). In contrast, the field
K
((
x
))
0
of all formal power series over
K
that converge at some element of
K
is algebraically closed in
K
((
x
)) (Proposition 2.4.5).