الفهرس 
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Abstract
The extensions of rings provide a useful tool for obtain-
ing algebraic theorems and results and additional struc-
tures for both rings, prime ideals of both rings, modules
over both rings with some functors and algebras over both
rings. The second level of prime ideals; in general of ideal
theory, occupies an important place in the algebraic geom-
etry.
Ring extensions, like field extensions, can be considered
from two points of view. One can look upward from a ring
to its extensions or downward to its subrings. The work
in this thesis provides an example of the upward point of
view.
There are many different ways of constructing the ring
extensions and their applications; several of these were
thought to be different in some papers, see [5], [9], [11],
[14], [17], [18], [25] and [26].
Graded ring extensions permit us to construct struc-
ture sheaves extensions, on the micro-structure sheaves,
on the graded prime spectrum
Spec
g
(
G
(
R
)) of the associ-
ated graded ring
G
(
R
), when this space is endowed with
the Zariski graded topology.
This thesis is devoted to the study of filtered and graded
Procesi extensions of filtered and graded rings. After a
ii
brief study of general features concerning filtered and graded
Procesi extensions at the level of graded Rees rings and
their associated graded rings, we turn to the behavior of
filtered and graded Procesi extensions towards the micro-
affine schemes. We show that these extensions behave well
from the geometric point of view.
The thesis consists of three chapters :
Chapter 1 : Preliminaries
This chapter provides the preliminaries and the back
ground material to be used in subsequent chapters. We
provide a brief survey of the basic definitions and elemen-
tary results concerning extensions of a ring, prime spec-
trum of a ring, graded rings and filtered rings as well as
sheaves and schemes.
Chapter 2 : Filtered and Graded Procesi Exten-
sions of Rings
In this chapter we continue the study of filtered and
graded Procesi extensions of filtered and graded rings in-
troduced in [17].
In the first section of this chapter, we define a new filtra-
tion
F
′′
S
of
S
:
F
′′
S
=
{
F
′′
n
S
}
n
∈
Z
; with
F
′′
n
S
=
φ
(
F
n
R
)
.S
R
.
As in [17], we study, over the filtered level, the passage of
iii
various ring theoretic properties between
R,S
and concern
with the relationship between the filtration
F
′′
S
on
S
and
those studied in [17].
In the second section, with respect to
F
′′
S
, we prove
that
G
(
S
) is graded Procesi extension of
G
(
R
) and, for
any
n
∈
Z
,
̃
S/X
n
̃
S
=
̄
̃
S
(
n
) is a graded Procesi extension
of
̄
̃
R
as in Proposition 2.2.5.
Chapter 3 : Procesi Extensions of Filtered and
Graded Rings Applied to the Micro-Affine Schemes
In this chapter we turn to the behavior and graded Pro-
cesi extensions towards to the micro-affine schemes . A
number of results concerning these concepts are given.
In the first two sections we introduce a survey, some-
times with proofs, on the graded spectrum of the asso-
ciated graded ring
G
(
R
) of
R
. Some results about the
micro-affine schemes are concerned, see [6], [10], [15],
[21] and [26]. This survey represents a solid foundation
for our results in this chapter.
In the third section we prove that the filtered Procesi
extension
φ
:
R
→
S
, for every
n
∈
Z
, induces a graded
Procesi extension
(
Y
=
Spec
g
(
G
(
S
))
,
̃
O
(
n
)
Y
)
−→
(
X
=
Spec
g
(
G
(
R
))
,
̃
O
(
n
)
X
)
of affine schemes.
iv
By considering the inverse limit in the graded sense and
the idea of micro-affine structure sheaves, we pay our at-
tention to deduce that
(
Y,O
μ
Y
)
→
(
X,O
μ
X
)
, of filtered micro-affine schemes, is a filtered Procesi ex-
tension.
The main results of Chapters 2 and 3
seem to
be original and have been published in the International
Mathematical Forum (Bulgaria), Vol.7, 2012, no. 26, 1279
-1288.