Talk:Endomorphism ring

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Field: Algebra

Notation

The notation used in this article, End(V), to denote the set of all homomorphisms of an abelian group V into itself, is common in many algebra texts. The notation, Hom(V,W) is also common in many algebra texts, when V and W are two separate abelian groups. Donald S. Passman defines the notation as, "Let V and W be additive abelian groups. ... The set of all such homomorphisms is denoted by Hom(V,W). ... When W = V, we call $\alpha$:VV an endomorphism of V and write End(V) = Hom(V,V)." I have only seen one text, written by John B. Fraleigh, use Hom(V) for the set of endomorphisms. Anita5192 (talk) 04:17, 27 July 2012 (UTC)

OK I'm glad we have this information. Every once in a while, an author comes up with a term that never gets used (like Lang's "entire ring" for "domain".) I haven't ever had a chance to see Fraleigh and now I'm curious about it. Rschwieb (talk) 16:37, 27 July 2012 (UTC)

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)