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Friday, July 24, 2020 | History

2 edition of Some lattice theoretic theorems concerning the submodules of a module found in the catalog.

# Some lattice theoretic theorems concerning the submodules of a module

## by Gary Dean Jensen

Written in English

Subjects:
• Mathematics

• ID Numbers
Open LibraryOL25311939M

Serial modules. Since ancient times, the modules whose submodule lattice is a chain, the so called serial (or uniserial) modules, have attracted a lot of in-terest, these are those modules M such that any pair N1,N2 of submodules is comparable: we have N1 ⊆N2 or N2 ⊂N1. First of all, for some quite important rings, all the indecomposable. History. The isomorphism theorems were formulated in some generality for homomorphisms of modules by Emmy Noether in her paper Abstrakter Aufbau der Idealtheorie in algebraischen Zahl- und Funktionenkörpern which was published in in Mathematische general versions of these theorems can be found in work of Richard Dedekind and previous papers by .

We define some purely lattice theoretic translations of algebraic notions related to submodule lattices, leading to new structural features of modular . We prove the subset of a module consists of elements annihilated by some power of a given ideal is a submodule. We construct an ascending chain of submodules.

They expand the discussion by exploring advanced theorems and new classes, such as new chain conditions, TS-module theory, and the lattice of prenatural classes of right R-modules, which contains many of the previously used lattices of module classes. The book finishes with a study of the Boolean ideal lattice of a ring. Formal definition. Let R be an integral domain with field of fractions R-submodule M of a K-vector space V is a lattice if M is finitely generated over R and R-torsion-free (no non-zero element of M is annihilated by a non-zero element of R).It is full if V = KM.. Pure sublattices. An R-submodule N of M that is itself a lattice is an R-pure sublattice if M/N is R-torsion-free.

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### Some lattice theoretic theorems concerning the submodules of a module by Gary Dean Jensen Download PDF EPUB FB2

Theorem If M is an A-R module, which satisfies the A.C.C. for A submodules and R submodules, then the rings A/q and R/a' satisfy the A.C.C. for right ideals, where q and q' are the annihilating ideals of M in A and R respectively. Proof. We shall show that the A.C.C.

holds for right ideals in ^4/q. Enter the password to open this PDF file: Cancel OK. File name:. Some lattice theoretic theorems concerning the submodules of a module.

By Gary : Gary Dean. Jensen. SOME LATTICE THEORETIC THEOREMS CONCERNING THE SUBMODULES OF A MODULE SOME LATTICE THEORETIC THEOREMS CONCERNING THE SUBMODULES OF A MODULE. By and Gary Dean JensenThesi S and Gary Dean Jensen.

Abstract. Some lattice theoretic theorems concerning the submodules of a module Year: OAI identifier:. Let be a submodule. Then since since we have an Abelian group and further due to closedness under the module operation, also.

If is such that, then for any also. Definition and theorem (factor modules): If is a submodule of, the factor module by is defined as the group together with the module operation.

In some situations the lattice of submodules Λ of a module M can b e deter- mined completely. Indeed if M is multiplicity-free, then Λ is distributive, so by theAuthor: Ian Musson. We denote by Lat(M) the lattice of all submodules of a module M. A module is distributive if F N (G + H) --F n G + F n H for all F, G, H 6 Lat(M).

A module M is uniserial if any two submodules of M are comparable with respect to inclusion. All uniserial modules are : Askar A. Tuganbaev. module M is said to be indecomposable if it has no proper nontrivial com- Simple and Semisimple Rings and Modules plemented submodule M 1, i.e., if M = M 1 ' M 2 implies that M 1 = h 0 i orFile Size: KB.

Model theory of modules Consequences are: Every module is elementarily equivalent to a direct sum of indecomposables. (1) the elementary type of the module M is determined by the invariants I,(M) = min?‘L(U) log P/+(M) 1 cp, 4 ppfl (UEUJ). For the computation of the IU it is enough to let the pairs cp, C/J range over a.

module is obtained essentially by a modest generalisation of that of a vector space, it is not surprising that it plays an important role in the theory of linear algebra.

Modules are also of great importance in the higher reaches of group theory and ring theory, and are fundamental to the study of advanced topics such as homological.

A module M is uniserial if any two submodules of M are comparable with respect to inclusion. All uniserial modules are distributive.

Any quasicyclic Abelian p-group is a uniserial module. LIBRARY NAVALPOSTGRADUATESCHOOL PREFACE Thepurposeofthispaperistoinvestigatecer-tainpropertiesofthelatticeofsubmodulesofamodule. In the branch of mathematics called order theory, a modular lattice is a lattice that satisfies the following self-dual condition: Modular law a ≤ b implies a ∨ = ∧ b for every x, where ≤ is the partial order, and ∨ and ∧ are the operations of the lattice.

This phrasing emphasizes an interpretation in terms of projection onto the sublattice, a fact known as the diamond isomorphism theorem. The lattice of submodules of any module over any ring is modular and complete. The lattice of a finitely generated/Noetherian module needs not be distributive.

Example: Let F be a field (finite, if you like) and M = F × F be a vector space of dimension 2. The lattice of submodules has the diamond lattice. Cȃlugȃreanu [2] used lattice theory in module theory and studied several concepts from module theory in lattice theory.

Keskin [11] obtained some properties of extending modules using modular Author: Grigore Călugăreanu. Topics in Lattice Theory with Applications to Rings, Modules and Categories Lecture Notes, Escola de Algebra, XXIII Brazilian Algebra Meeting, Maringa, Parana, Brasil,80 pages. Book Author: Toma Albu.

HYPERRADICAL - HOPKINS–LEVITZKI THEOREM 3 Deﬁnition 2. A lattice L is modularif it satisﬁes x ≤ z ⇒ x ∨(y ∧z) = (x ∨y)∧ z for any x,y,z ∈ L. If M is a left-module over a ring R, the lattice Sub(M) of submodules of M is a modular lattice. In particular, the lattice of left-ideals of R is modular.

Deﬁnition 3. sending (B, A)-submodules onto ideals of B. More generally,if U is any finitely generated projective faithful left B -module over any ring B, and A = End B U then there is a lattice isomorphism { r i g h t A − s u b m o d u l e s o f U → (r i g h t i d e a s o f V) T I U ↔ I = I TCited by: The set of submodules of a given module M, together with the two binary operations + and ∩, forms a lattice which satisfies the modular law: Given submodules U, N 1, N 2 of M such that N 1 ⊂ N 2, then the following two submodules are equal: (N 1 + U) ∩ N 2 = N 1 + (U ∩ N 2).

A lattice $(L,\leq)$ is said to be modular when $$(\forall x,a,b\in L)\quad x \leq b \implies x \vee (a \wedge b) = (x \vee a) \wedge b,$$ where $\vee$ is the join operation, and $\wedge$ is the meet operation. (Join and meet.)The ideals of a ring form a modular lattice.

So do submodules of a module. These facts are easy to prove, but I have never seen any striking examples of their. A submodule of a finitely generated free module over a principal ideal domain is free. (For example, we can use the structure theorem for finitely generated modules over principal ideal domains.) In particular, we need to think about commutative rings that are not principal ideal domains in order to answer your question.in lattice theory just for two years between andmany lattice theorists, including Gr atzer [27, p.

], say that his results belong to the deepest part of lattice theory. For instance, Birkho, the founder and pi-oneer of universal algebra and lattice theory, wrote in .In this paper we introduce a lattice structure as a generalization of meet-continuous lattices and quantales.

We develop a point-free approach to these new lattices and apply these results to particular, we give a module counterpart of the well known result that in a commutative ring the set of semiprime ideals, that is, radical ideals is a by: 4.