Is a finitely generated abelian group?

Stated differently the fundamental theorem says that a finitely generated abelian group is the direct sum of a free abelian group of finite rank and a finite abelian group, each of those being unique up to isomorphism. The finite abelian group is just the torsion subgroup of G.

What is meant by finitely generated abelian groups?

Definition. A group G is torsion free if the only element of finite order is the identity. Definition. An abelian group G is finitely generated if there are elements such that every element can be written as. Note that this expression need not be unique.

How do you classify abelian groups?

Abelian groups can be classified by their order (the number of elements in the group) as the direct sum of cyclic groups. More specifically, Kronecker’s decomposition theorem. An abelian group of order n n n can be written in the form Z k 1 ⊕ Z k 2 ⊕ …

Are finitely generated abelian groups cyclic?

The fundamental theorem of finitely generated abelian groups tells us that every finitely generated abelian group is isomorphic to a direct product of cyclic groups. Cyclic groups are groups that can be generated by just one element.

How many finite Abelian groups are there?

The Fundamental Theorem of Finite Abelian Groups tells us that we have the following six possibilities.

What is the meaning of finitely generated?

In algebra, a finitely generated group is a group G that has some finite generating set S so that every element of G can be written as the combination (under the group operation) of finitely many elements of the finite set S and of inverses of such elements.

What is finitely generated K algebra?

In mathematics, a finitely generated algebra (also called an algebra of finite type) is a commutative associative algebra A over a field K where there exists a finite set of elements a1,…,an of A such that every element of A can be expressed as a polynomial in a1,…,an, with coefficients in K.

Are submodules of finitely generated modules finitely generated?

Some facts. Every homomorphic image of a finitely generated module is finitely generated. In general, submodules of finitely generated modules need not be finitely generated. As an example, consider the ring R = Z[X1, X2, …] of all polynomials in countably many variables.

What are the properties of an Abelian group?

Abelian Group

So, a group holds five properties simultaneously – i) Closure, ii) Associative, iii) Identity element, iv) Inverse element, v) Commutative.

What is direct sum in group theory?

The group direct sum of a sequence of groups is the set of all sequences , where each is an element of , and is equal to the identity element of for all but a finite set of indices . It is denoted. (1) and it is a group with respect to the componentwise operation derived from the operations of the groups .

Is free module finitely generated?

A finitely generated torsion-free module of a commutative PID is free. A finitely generated Z-module is free if and only if it is flat. See local ring, perfect ring and Dedekind ring.

Is a Noetherian ring finitely generated?

A ring R is Noetherian if any ideal of R is finitely generated. This is clearly equivalent to the ascending chain condition for ideals of R. By Lemma 10.28. 10 it suffices to check that every prime ideal of R is finitely generated.

Is every Artinian module Noetherian?

Since an Artinian ring is also a Noetherian ring, and finitely-generated modules over a Noetherian ring are Noetherian, it is true that for an Artinian ring R, any finitely-generated R-module is both Noetherian and Artinian, and is said to be of finite length; however, if R is not Artinian, or if M is not finitely- …

Does every module have a basis?

It is well known that a vector space V is having a basis, i.e. a subset of linearly independent vectors that spans V. Unlike for a vector space, a module doesn’t always have a basis.

Is every free module projective?

Every free module is projective. Proof. Let F be a free module over a ring R with basis B = {xi | i ∈ I }. If ψ: M → N is an R-module epimorphism and f ∈ HomR(F, N), then to each f(xi) (i ∈ I), there exists yi ∈ M so that ψ(yi) = f(xi) for all i ∈ I.

What is the dimension of module?

In abstract algebra, the length of a module is a generalization of the dimension of a vector space which measures its size. In particular, as in the case of vector spaces, the only modules of finite length are finitely generated modules. It is defined to be the length of the longest chain of submodules.

Is a ring a module?

In mathematics, a module is a generalization of the notion of vector space in which the field of scalars is replaced by a ring. The concept of module generalizes also the notion of abelian group, since the abelian groups are exactly the modules over the ring of integers.

What is another word for module?

In this page you can discover 16 synonyms, antonyms, idiomatic expressions, and related words for module, like: modules, faculty, program, coursework, half-unit, short-course, course, component, mental faculty, tutorial and assignment.