Sage-Tips

Is every abelian group a ring?

An important exception though, is that every finitely generated abelian group admits a ring structure with 1. (This follows from the classification of finitely generated abelian groups as direct products of cyclic groups).

Is every abelian group AZ module?

Any abelian group is a Z-module, where the action of Z is defined by na := a + ··· + a (n-fold sum). Every abelian group is a Z module in a unique way, and every homomorphism of abelian groups is a Z-module homomorphism in a unique way. Example. Just as any field F is vector space over F, any ring R is an R-module.

Is every abelian group is cyclic?

All cyclic groups are Abelian, but an Abelian group is not necessarily cyclic. All subgroups of an Abelian group are normal. In an Abelian group, each element is in a conjugacy class by itself, and the character table involves powers of a single element known as a group generator.

Is G an abelian group justify your answer?

If every element of a group is its own inverse, then show that the group must be abelian . (a * b ) = b * a ( Since each element of G is its own inverse) Hence, G is abelian.

Is a module a ring?

In a vector space, the set of scalars is a field and acts on the vectors by scalar multiplication, subject to certain axioms such as the distributive law. In a module, the scalars need only be a ring, so the module concept represents a significant generalization.

Is a polynomial ring a module?

The polynomial ring R[X] is a free R-module with basis 1, X, X2,… .

Is every Abelian group the additive group of some ideal in some ring?

No, there are many abelian groups such that only zero multiplication can be defined over them. One more interesting problem is to determine which subgroups of an abelina group A can be realized as ideals in some ring with additive group A.

What do you mean by an Abelian group if a group G has four elements then prove that it must be Abelian?

Answer:All elements in such a group have order 1,2 or 4. If there’s an element with order 4, we have a cyclic group – which is abelian. Otherwise, all elements ≠e have order 2, hence there are distinct elements a,b,c such that {e,a,b,c}=G.