APS of rings

From Apstheory

Current revision as of 03:00, 30 December 2006

An APS of rings is an APS over the category of rings. More specifically, an APS Failed to parse (Can't write to or create math temp directory): (G,\\Phi)

of rings is the following data:

• For each natural number Failed to parse (Can't write to or create math temp directory): n

, a ring, denoted Failed to parse (Can't write to or create math temp directory): G_n .

• For each ordered pair Failed to parse (Can't write to or create math temp directory): (m,n)

of natural numbers, a homomorphism Failed to parse (Can't write to or create math temp directory): \\Phi_{m,n}:G_m
× Failed to parse (Can't write to or create math temp directory): G_n
→ Failed to parse (Can't write to or create math temp directory): G_{m+n}

.

Satisfying the following compatibility conditions:

For Failed to parse (Can't write to or create math temp directory): g, h, k

in Failed to parse (Can't write to or create math temp directory): G_m, G_n, G_p
respectively, 

Failed to parse (Can't write to or create math temp directory): \\Phi_{m+n,p} (\\Phi_{m,n}(g,h),k) = \\Phi_{m,n+p} (g, \\Phi_{n,p}(h,k)) .

The above condition is termed an associativity condition.

We may assume Failed to parse (Can't write to or create math temp directory): G_0

as the trivial ring and define Failed to parse (Can't write to or create math temp directory): \\Phi_{m,0}
and Failed to parse (Can't write to or create math temp directory): \\Phi_{0,n}
as trivial paddings.

[edit] Terminology

[edit] Members and elements

For an APS Failed to parse (Can't write to or create math temp directory): (G,\\Phi)

the member Failed to parse (Can't write to or create math temp directory): G_n
is termed the Failed to parse (Can't write to or create math temp directory): n^{th}
member of the APS. A member of the APS is an object that is the Failed to parse (Can't write to or create math temp directory): n^{th}
member for some Failed to parse (Can't write to or create math temp directory): n

.

An element of the APS is an element of some member of the APS.

The home of an element of the APS is the member in which it lies. The index of a member is the Failed to parse (Can't write to or create math temp directory): n

for which it is the Failed to parse (Can't write to or create math temp directory): n^{th}
member.

[edit] Block concatenation map

The maps Failed to parse (Can't write to or create math temp directory): \\Phi_{m,n}

are termed block concatenation maps.

[edit] Ground member

The ground member of an APS is its first member.

[edit] Other notions

[edit] Homomorphism of APSes

{{further|APS homomorphism}

Given APSes Failed to parse (Can't write to or create math temp directory): (G,\\Phi)

and Failed to parse (Can't write to or create math temp directory): (H,\\Psi)

, a homomorphism Failed to parse (Can't write to or create math temp directory): h: G

→ Failed to parse (Can't write to or create math temp directory): H
associates, to each natural number Failed to parse (Can't write to or create math temp directory): n

, a map Failed to parse (Can't write to or create math temp directory): h_n: G_n

→ Failed to parse (Can't write to or create math temp directory): H_n

, such that:

Failed to parse (Can't write to or create math temp directory): \\Psi_{m,n}(h_m(a),h_n(b)) = h_{m+n}(\\Phi_{m,n}(a,b))

[edit] Sub-APS notion

Further information: sub-APS

Given an APS Failed to parse (Can't write to or create math temp directory): (G,\\Phi) , a sub-APS Failed to parse (Can't write to or create math temp directory): H

associates, to each Failed to parse (Can't write to or create math temp directory): n

, a subring Failed to parse (Can't write to or create math temp directory): H_n

of Failed to parse (Can't write to or create math temp directory): G_n

, such that the image of Failed to parse (Can't write to or create math temp directory): H_m

× Failed to parse (Can't write to or create math temp directory): H_n
under Failed to parse (Can't write to or create math temp directory): \\Phi_{m,n}
lies inside Failed to parse (Can't write to or create math temp directory): H_{m+n}

.

When the APS of groups is injective, any sub-APS is also injective.

[edit] Quotient APS notion

Further information: quotient APS

A quotient APS is the image of an APS in a homomorphism that is surjective at each member.

[edit] Normal sub-APSes, kernels and images

Given a homomorphism of APSes of rings, the kernels of the individual homomorphisms form a set-theoretic sub-APS, which is in fact an ideal APS in the APS of rings.

Further, we have the following result: a set-theoretic sub-APS of an APS of groups occurs as the kernel of an APS homomorphism if and only if every member of it is a two-sided ideal of the corresponding member of the whole APS. A sub-APS satisfying either of these equivalent conditions is termed a two-sided ideal APS.

This parallels the ring theory result that a subset of a ring is the kernel of a ring homomorphism from it if and only if it is a two-sided ideal.

[edit] Properties

[edit] Injectivity

An APS of rings is termed injective, or an IAPS of rings, if every block concatenation map is injective. For an IAPS of rings, we usually also assume the condition of refinability.

[edit] Commutativity

Very few APSes of rings are commutative. Note that a commutative APS cannot also be injective.

[edit] Padding-injectivity

Most APSes of rings that we encounter satisfy the condition of being padding-injective.