Thread: Direct Products and SUms

Hi,

If I have two algebraic objects (sets, groups, rings, modules etc) and call them A and B can someone please give a definition of the following operations?

+ addition
x product
- direct sum
- direct product

Cheers

well, yes, but you may not like it:

the direct sum (or "co-product") A₁+A₂ of two (or more) objects A₁,A₂ is ANY object (up to a (____)-isomorphism) that comes with two (or more...depending on how many "factors") (usually, but in some structures not necessarily, injective) morphisms:

i₁:A₁-->A₁+A₂, i₂:A₂-->A₁+A₂

such that if X is any other object (in the same structure-type) with two (or more...) morphisms f₁:A₁-->X, f₂:A₂-->X

there exists a *unique* morphism f:A₁+A₂-->X with:

f₁ = foi₁
f₂ = foi₂

(often this is phrased as: an indexed set of morphisms from the factors to X factor through the "canonical injections" (the map f in the case of two factors is often written "f₁+f₂")).

the direct product (or "product") is the "dual" construction, now we have morphisms p₁:A₁xA₂-->A₁, p₂:A₁xA₂-->A₂ (usually, but not necessarily surjective), and if f₁:X--->A₁, and f₂:X--->A₂ then there is a morphism f (often called f₁xf₂) which is unique, such that:

f₁ = p₁of
f₂ = p₂of

the maps p_j are often called "canonical projections onto the factors" and we say the maps f_j "factor through the projections".

these constructions can vary widely, depending on the structure underlying our objects:

SETS: coproduct = disjoint union, product = cartesian product
GROUPS: coproduct = free product, product = direct product
ABELIAN GROUPS : coproduct = direct sum, product = direct product (for finite abelian groups, these are the same, but for an infinite number of factors of abelian groups the direct product is "bigger")
VECTOR SPACES/MODULES: coproduct = direct sum, product = direct product (for finite-dimensional vector spaces, these are the same. for modules, the same considerations on the number of factors apply as for abelian groups).


with the most common algebraic objects, the canonical injections can be thought of as inclusion maps, which are injective homomorphisms, and the canonical projections can be thought of as quotient maps, which are surjective homomorphisms. in many cases, the direct sum and the direct product coincide, and the terms are used interchangeably:

Z + Z = Z x Z

here is one example, to help you understand the difference:

the additive group (R[x],+) (the additive group of all real polynomials) is (isomorphic to) the direct sum of countably many copies of (R,+)

the additive group (R[[x]],+) (the additive group of all real formal power series in x) is (isomorphic to) the direct product of countably many copies of (R,+).

it is a worth-while exercise to show that the usual definition of direct product/sum satisfy these so-called "universal properties" (doing so will also indicate why if something does satisfy them, it is isomorphic to a direct product/sum).