Talk:Endomorphism ring

Properties

The last property states:

• The formation of endomorphism rings can be viewed as a functor from the category of abelian groups (Ab) to the category of rings.

Is this really true? What does End do with morphisms?

E.g. for the trivial morphism g : G -> H what should h = End(g)(f) for f : G -> G look like? The only requirement I see is g o f = h o g but this holds for any h since g is trivial.

If it's true some more information would be great since it seems not entirely obvious.

bastian 153.96.12.26 (talk) 12:05, 29 November 2012 (UTC)

I agree with you and I removed it. If you would have asked me "can it be viewed as a functor" today, I think I would have said "no," but nevertheless it looks like I was responsible for adding this earlier this year! It looks like I was expanding the section with the goal of showing connections between the module and endomorphism ring, and this looks like something I cooked up in that fervor. Thanks for catching it! Rschwieb (talk) 15:14, 29 November 2012 (UTC)

Elaboration of definition seems needed

The following sentence might bear elaboration, possibly in a definition section:

The addition operation is defined by pointwise addition of functions and the multiplication operation is defined by function composition.

In particular:

• that "functions" refers specifically to the elements of the endomorphism ring is not abundantly clear; it has to be inferred
• that "pointwise addition of functions" uses the group operation as the "addition" operation on the domain for the "pointwise" operation is not immediately clear; this relies on the reader being familiar with the group operation is typically called addition rather than multiplication in the terminology of abelian groups. It is easily first assumed that the group operation should map to the composition operation (as in Cayley's theorem), which leaves one initially wondering what "addition" is.

— Quondum 16:55, 11 August 2013 (UTC)

Good ideas. I went ahead and made such changes to the Description section, but not the lead, for the sake of brevity. Rschwieb (talk) 17:01, 14 August 2013 (UTC)

I've tweaked it a bit more. Feel free to crit/modify what I've done. — Quondum 13:13, 15 August 2013 (UTC)

Constructing of an endomorphism algebra

From the lead:

As the resulting object is often an algebra over some ring R, this may also be called the endomorphism algebra.

It strikes me that this is ever so slightly pulled out of the air. If the endomorphism ring is of some object _RM in the category of left R-modules, it might be natural to extend the definition of the endomorphism ring to a left R-algebra (does this make sense?) by defining pointwise that (rf)(x) := rf(x) for all r ∈ R and x ∈ _RM. This should be clarified in the Description section. — Quondum 22:56, 18 August 2013 (UTC)

Zero map

How can the zero map ${\textstyle 0:x\mapsto 0}$ be in the ring when it's not an endomorphism? Luke Maurer (talk) 00:03, 27 February 2019 (UTC)

The zero map is indeed an endomorphism. The zero map maps all elements of G to the identity element, 0. Since this maps every element of G into G, this is an endomorphism.—Anita5192 (talk) 00:14, 27 February 2019 (UTC)

Smoother lead section?

IMHO, the lead section currently reads as if there were a missing paragraph. The first paragraph talks about endomorphism rings of abelian groups. The second paragraph suddenly jumps into category theory in a way that actually made me look in the article's history for signs of vandalism. Can somebody knowledgeable please try to smoothen the thematic gap between these paragraphs? – Tea2min (talk) 20:41, 27 December 2020 (UTC)