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Math and science::Algebra::Aluffi

Ring. As a group of group endomorphisms.

A fundamental way to motivate rings is via endomorphisms of a group.

A requirement of this construction is that the original group be abelian. The reason is captured below.

Setup

  • \( G \) is a group.
  • \( \groupAdd{G} : G \times G \to G \) is the group operation of \( G \).
  • \( \rho, \phi \in \operatorname{Endo}(G) \) are group endomorphisms.
  • \( + : \operatorname{Endo}(G) \times \operatorname{Endo}(G) \to \operatorname{Endo}(G) \) is the derived group operation of the ring.
  • \( a, b \in G \) are two elements of \( G \) and \( x = a \groupAdd{G} b \).

Requirement for \( + \)

The term \( \rho{\small{+}}\phi(x) \) decomposes in two ways, which must be equal.

Firstly, by definition of the new operation:

\[ \begin{align*} \rho{\small{+} }\phi(x) &= \rho(x) \groupAdd{G} \phi(x) \\ &= \rho(a) \groupAdd{G} \rho(b) \groupAdd{G} \phi(a) \groupAdd{G} \phi(b) \end{align*} \]

And as \( + \) produces a group homomorphism:

\[ \begin{align*} \rho{\small{+} }\phi(x) &= \rho{\small{+} }\phi(a) \groupAdd{G} \rho{\small{+} }\phi(b) \\ &= \rho(a) \groupAdd{G} \phi(a) \groupAdd{G} \rho(b) \groupAdd{G} \phi(b) \\ \\ \end{align*} \]

The requirement for these to decompositions to be equal is what restricts the construction to abelian groups.


From groupoid to ring

Zooming in and then zooming out.

Group as a type of groupoid

Recall that a group can be considered a groupoid with a single object. A groupoid being a category where every morphism is an isomorphism. An isomorphism being a homomorphism with an inverse.

From this perspective, if a group were considered a set, then the elements would be the endomorphisms.

Category of groups

Zooming out, now consider the category of groups. An object of the above category is gathered up with all of it's endomorphisms to form an object in the category of groups. The homomorphisms between objects are now group homomorphisms. Group homomorphisms are relatively complex in how they must preserve group structure.

Group of group endomorphisms

Now take a single group object and consider it's group endomorphisms. It will have at least the function that sends all elements to the identity, and it will have the identity function, which will be different to the zero function if there is more than one element.

We can create a new composition rule-table by inheriting the behavior from the underlying group. Let \(m_G : G \times G \to G \) represent the group operation of a group \(G \). We will create another operation \( m_E : \operatorname{Endo}(G) \to \operatorname{Endo}(G) \). Let \( \rho, \phi \in \operatorname{Endo}(G) \). Then \( m_E \) is defined as follows:

\[ m_E(\rho, \phi)(t) = m_G(\rho(t), \phi(t)) \]

In alternative syntax, we write:

\[ \rho{+}\phi(t) = \rho(t) + \phi(t) \]

where the RHS \( + \) is the original group operation, and the LHS \( + \) is just syntax.

This operation is a group operation iff the underlying group is abelian.

The second operation

To form a ring, we need an additional operation that is associative. As the elements of our new group are themselves homomorphisms, they already have an associative composition operation (function composition). This operation forms the multiplication operation of the ring. It can be checked that it distributes with the first group operation.