## Endomorphism Ring of an Abelian Group

**Point of Post: **In this post we discuss the endomorphism ring of an abelian group as being a ‘typical ring’ in the sense that every ring is embeddable into the endomorphism ring of its underlying group structure.

*Motivation*

This post will inaugurate a short run of posts developing certain classes of rings (endomorphism rings, rings of continuous functions, Boolean rings, ring of ring morphisms, matrix rings, group rings, etc.) to give us a firm set of examples. Our first class of rings will be endomorphism rings of abelian groups. Intuitively how one would think to create the endomorphism ring of an abelian group would be to note that for a given abelian group and endomorphisms there are two ‘natural’ operations to perform on them, namely you can add them (in the sense that ) and you can compose them . It turns that these two operation on the set of all endomorphisms on form a ring. That said, unlike the other aforementioned rings which clearly come up often in mathematics the endomorphism ring of an abelian group seems a bit…well, contrived. So, why would anyone really care about it at this stage in our studies? Well, the main reason why the endomorphism ring of an abelian group is interesting is it lets us formulate a unital ring theoretic analogy to the group theoretic Cayley’s theorem. Namely, we shall see that every ring naturally embeds into the endomorphism ring of its underlying group structure.

*Endomorphism Ring of an Abelian Group*

Let be an abelian group. Let denote the set of all group endomorphism of . Then if we define the *sum * of to be such that for every and multiplication to be the composition then forms a unital ring. To see this we note that evidently and form an abelian group (one can check this is where you need that is abelian) and monoid (with unit equal to the identity map ) respectively and thus it suffices to show distributivity. But, this is clear since, for example, given any and we have that

For a specific example we consider the following:

**Theorem: ***Let then .*

**Proof: **Define given by where, for the sake of notational convenience, we denote by . We first note that is well-defined (in the sense that ) by right distributivity. To see that is a morphism we merely note that for any

and

so that and and so is a morphism as claimed. Clearly is injective since if then . To see that is surjective we merely note that if then for any we have that and so by definition and so in particular .

This example motivates the following unital ring-theoretic analogue of Cayley’s theorem:

**Theorem: ***Every unital ring is canonically isomorphic to a subring of for some abelian group .*

**Proof: **Consider (considering only ‘s underlying group structure) and define given by . Using the ring structure similar to the previous theorem it’s easy to see that is a morphism. Moreover, using the fact that for every we see that is injective. Thus, we have that and is a subring of .

It should be seen equally as easy that the above in fact generalizes to

**Theorem: ***Let be a ring (not necessarily unital) with at least one non-zero divisor. Then, is canonically isomorphic to a subring of .*

**Proof: **It evidently suffices to prove injectivity. To do this we merely note that implies for all . But, if is the guaranteed non-zero divisor then implies and so as claimed.

**References:**

1. Dummit, David Steven., and Richard M. Foote. *Abstract Algebra*. Hoboken, NJ: Wiley, 2004. Print.

2. Bhattacharya, P. B., S. K. Jain, and S. R. Nagpaul. *Basic Abstract Algebra*. Cambridge [Cambridgeshire: Cambridge UP, 1986. Print.

Thank you for this post. It has the information necessary to introduce very quickly the Endomorphism Ring without distracting examples

Comment by jocerfranquiz | August 12, 2011 |

You’re welcome, I’m glad it helped! Thank you for caring enough to respond!

Comment by Alex Youcis | August 12, 2011 |

[…] know as much ring theory) is that, by definition, for an abelian group , where is the endomorphism ring but it is easy to show that if is […]

Pingback by The Unit and Automorphism Group of Z/nZ (pt. II) « Abstract Nonsense | September 10, 2011 |

[…] something that fits the bill perfectly. In particular, if happened to be an abelian group then we know that set is naturally a ring under composition and point-wise addition. Thus, what if we defined […]

Pingback by Modules (Pt. I) « Abstract Nonsense | October 27, 2011 |

[…] . The set of all such endomorphisms will be denoted . Now, by virtue of being an abelian group we know that is a ring under pointwise addition and composition of functions. But, we know that if we […]

Pingback by Homomorphism Group (Module) of Two Modules (Pt. II) « Abstract Nonsense | November 1, 2011 |

[…] Consider the functor which sends to , the endomorphism ring of the group. […]

Pingback by Functors (Pt. I) « Abstract Nonsense | December 27, 2011 |

[…] as nothing more than an abelian group and a unital ring homomorphism (where, as usual, is the endomorphism ring)–but from first principles this is nothing more than a left (or right) -module. This examples […]

Pingback by Ab-categories and Preadditive Categories (Pt. II) « Abstract Nonsense | January 2, 2012 |

[…] theory of finite groups. As a last example, let’s consider unital rings. We know then that every ring embeds into the endomorphism group of an abelian group. Thus, the above […]

Pingback by Yoneda’s Lemma (Pt. I) « Abstract Nonsense | January 14, 2012 |