A characterization of tame automorphisms of the algebra
A
=
F
x
1
,
x
2
,
x
3
A=Fx_1,x_2,x_3
of polynomials in three variables over a field
F
F
of characteristic
0
0
is obtained. In particular, it ...is proved that the well-known Nagata automorphism is wild. It is also proved that the tame and the wild automorphisms of
A
A
are algorithmically recognizable.
We say that a commutative ring
R
satisfies the restricted minimum (RM) condition if for all essential ideals
I
in
R
, the factor
R
/
I
is an Artinian ring. We will focus on Noetherian reduced rings ...because in this setting known results for RM domains generalize well. However, as we will show, RM rings need not be Noetherian and may have nilpotent elements. One of the classic results in the theory of RM rings is that for Noetherian domains the RM condition corresponds to having Krull dimension at most one. We will show that this can be generalized to reduced Noetherian rings, thus proving that affine rings corresponding to curves are RM. We will give examples showing that the assumption that the ring is reduced is not superfluous. In the second part, we will study CDR domains, i.e., domains where for any two ideals
I
,
J
the inclusion
I
⊆
J
implies that
I
is a multiple of
J
. We will prove that CDR domains are RM and this will allow us to give a new characterization of Dedekind domains. Examples of RM rings for various classes of rings will be given. In particular, we will show that a ring of polynomials
R
x
is RM if and only if
R
is a reduced Artinian ring. And we will study the relation between RM rings and UFDs.
We introduce a Poisson bracket on the ring of polynomials
A
=
F
x
1
,
x
2
,
…
,
x
n
A=Fx_1,x_2, \ldots ,x_n
over a field
F
F
of characteristic
0
0
and apply it to the investigation of subalgebras ...of the algebra
A
A
. An analogue of the Bergman Centralizer Theorem is proved for the Poisson bracket in
A
A
. The main result is a lower estimate for the degrees of elements of subalgebras of
A
A
generated by so-called
∗
\ast
-reduced pairs of polynomials. The estimate involves a certain invariant of the pair which depends on the degrees of the generators and of their Poisson bracket. It yields, in particular, a new proof of the Jung theorem on the automorphisms of polynomials in two variables. Some relevant examples of two-generated subalgebras are given and some open problems are formulated.
Let
Ω
be an uncountable and algebraically closed field. We prove that every ideal of the polynomial ring
R
=
Ω
x
1
,
x
2
,
…
is the intersection of ideals of the form
{
f
∈
R
:
D
(
f
g
)
(
c
)
=
0
...for every
g
∈
R
}
, where
D
is a differential operator of locally finite order, and
c
is a vector with values in
Ω
.
Rings of Polynomials Evyatar, A.; Zaks, A.
Proceedings of the American Mathematical Society,
01/1970, Volume:
25, Issue:
3
Journal Article
Peer reviewed
Open access
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