The Book Deathwatch By Robb White

by the Group to the building

We now know from her previous post, what a group is.
Here there is a definition for the group, which sounds a bit different, but
is equivalent to the others.

just to avoid misunderstandings: This Muliplikation can also be an addition, and one can therefore also be 0.

An important property that can occur when magma is the Commutativity .
A magma M is called commutative, if true:
If M is commutative and a group, we call M abelian or an abelian group.

So far we only considered a set with a link. But we can assign a set and two links.

Let M be a set with two operations

also be covered by the two distributive, namely

Then
a half ring .

A ring is a semiring
in the (R, +) an abelian group. The neutral element of (R, +) is denoted by "0". In most cases, is also required that the ring is a unit element "1" contains so
a neutral element of multiplication. The ring is then unitary .
are we when we speak of struggle, always talking about unitary rings.

A ring is called a division ring (or division ring) if R is a ring and if he has any nonzero element is a multiplicative inverse.
This is equivalent to saying that the ring is without the zero, a group with respect to the Multiply.

that was not enough. Over algebraic structures can be re-defined algebraic structures.

Let R be a ring. Let M be an additive abelian group. There is an additional link

also may apply the following:

Then M is an R-module links.

A right R-module M is defined the same, but the ring operated from the right. That is, the picture is now of the form f
and r, s are then the points (1) - (4) always written right of x, y.

ring is called commutative if the Multiplication in this ring is commutative.
If R is a commutative ring, so there is no difference between left and right module.
We only need the conditions for the R-module links verified.
If M is a left R-module over a commutative ring R, then M is called an R-module .

The following definitions can be there, where it says "module", always "left module" or always use "right module.

Let M be an R-module and N is a nonempty subset of N. Then N is a submodule means of M if N itself is an R-module. We write for "N is submodule of M"
Let N be a subset of a module M, it means the smallest submodule of M containing N, N of the product. We write for the product of N Let P be a subset of M. P is linearly dependent , if a finite number of pairwise different
If this is not the case, P is linearly independent .

Let B be a linearly independent subset of a module M with
Then B is a base M.

Be Rel is a relation on a set M.
Rel is reflexive applies if:
Rel is antisymmetric applies if:
Rel is transitive when:
A relation is partial order if it is reflexive, antisymmetric and transitive.
is For example, the set inclusion
a partial order.
says this: A is a subset of B.
A set with a partial order is called a partially ordered set .

Let R be a division ring and M a left R-module. The partially ordered by set inclusion set of all submodules of M is called the projective geometry P (M).

Let P (M) is a projective geometry. A flag of P (M) is a chain of pairwise different submodules

A real flag is a flag in which neither {0} or M appears.

One can interpret flags as sets of sub-modules. These we can sort by inclusion. The set of all real flags of a projective geometry, provided with the set inclusion, we call
A Frame in a projective geometry P (M) is a set of points
The Apartment S (F) of the frame F consists of the real flags
The set of all real flags and the set of all apartments in P (M)
give the building of P (M).