توصيفگر ها :
مشبكه ضربي , ساختار جبري , مدول هاي مشبكه , تجزيه اوليه
چكيده فارسي :
در اين پايان نامه با استفاده از تئوري ايدآل مجرد و تئوري مدول هاي مشبكه به مطالعه و مشخصه سازي يك le-مدول روي حلقه هاي تعويض پذير مي پردازيم. يك le-مدول روي يك حلقه ي تعويض پذير R، يك مونوئيد مشبكه ي كامل مرتب (M;+;≤; e) با بزرگترين عضو e و عملي مدول گونه از R روي آن است. هم چنين ما توپولوژي زاريسكي روي Spec(M) را مورد مطالعه و بررسي قرار و برخي از نتايج مربوط به توپولوژي زاريسكي حلقه ها را به le-مدول ها تعميم مي دهيم. در كنار برخي نتايج ديگر نشان مي دهيم كه فضاي توپولوژيكي Spec(M) همبند است اگر و تنها اگر حلقه ي خارج قسمتي R ∕ Ann(M) شامل هيچ خودتواني به جز 0 و 1 نباشد. همچنين نشان مي دهيم مجموعه هاي باز توپولوژي زاريسكي حلقه ي خارج قسمتي R ∕ Ann(M) پايه اي از مجموعه هاي باز شبه- فشرده براي توپولوژي زاريسكي روي Spec(M) القا مي كند. در ادامه برخي خصوصيات معادل براي طيفي بودن Spec(M) ارائه مي دهيم و نشان مي دهيم هر زيرمجموعه ي بسته ي تجزيه ناپذير Spec(M) يك نقطه ي عام دارد.
در پايان عناصر اول و اوليه را در يك le-مدول معرفي كرده و سپس دو قضيه ي تجزيه ي يكتايي اوليه را براي يك عضو زيرمدول در يك le-مدول لاسكرين اثبات مي كنيم.
رده بندي موضوعي:.54B35. 13C05. 13C99. 06F25
واژگان كليدي:مشبكه ي ضربي، ساختار جبري، مدول هاي مشبكه، تجزيه اوليه
چكيده انگليسي :
This thesis is based on the following papers:
1) A. K. Bhuniya and M. Kumbhakar,Uniqueness of primary decomposition in Laskerian le-modules , Acta
Math. Hunga. 158(1) (2019) 202-215.
2) M. Kumbhakar and A. K. Bhuniya,On the prime spectrum of an le-module , J. Algebra and Appl. 20
(12), 2150220 (2021).
First, we will introduce the notion of le-modules over commutative rings. Then, we will to study and
characterize le-modules over commutative rings. An le-module M over a commutative ring R is a complete
lattice ordered monoid (M,+,≤, e) with the greatest element e and module like action of R on it. Our
motivation comes from abstract ideal theory and the theory of lattice modules, and with a desire to develop
an alternative abstract submodule theory. An le-module M over the ring R abstracts the set of all subsets of
a module over R and submodules are characterized as distinguished elements in M.
We also introduce prime and primary elements in an le-module, and then we introduce and characterizes
Zariski topology on the set Spec(M) of all prime submodule elements of M. In fact, we extend many results
on Zariski topology for modules over a ring to le-modules. We show that for any le-module M, the
topological space Spec(M) is connected if and only if R/Ann(M) contains no idempotent other than 0 and
1. It is shown that open sets in the Zariski topology for the quotient ring R/Ann(M) induce a base of quasicompact
open sets for the Zariski-topology on Spec(M). Also, every irreducible closed subset of Spec(M)
has a generic point. Besides, we prove a number of different equivalent characterizations for Spec(M) to be
a spectral space. Finally, we establish two uniqueness theorems for primary decomposition of a submodule
element in a Laskerian le-module.
The radical of a submodule element n of RM is defined to be an ideal of the ring R, and so associated
primes of n are prime ideals of R. Thus we found direct ways of interaction between rings and le-modules.
In Section 2, we give the definition of an le-module and introduce the notion of submodule elements in
le-modules. Also we prove a number of basic results characterizing submodule elements. Section 3 is
devoted to characterizing prime and primary submodule elements and We study the primary decomposition
of a submodule element and their uniqueness.
Theorem (1st uniqueness theorem).Let n be a submodule element of an le-module RM and assume that
n has a reduced primary decomposition n = q1 ^ q2 ^ ^ qr. Let P be a prime ideal of R. Then
P = Rad(qi) for some i if and only if (n : x) is a P-primary ideal of R for some x ≰ n. Hence the set of
all associated primes is independent of primary decomposition of n.
Theorem (2nd uniqueness theorem). Let RM be an le-module and n a submodule element of M. If
fPi1 ; Pi2 ; :::; Pi k
g is a set of isolated prime divisors of n then qi1
^ qi2
^ ^ qi k depends only on this
set and not on the particular reduced primary decomposition of n.
