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?
- direct sum
- direct product
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:
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 pj are often called "canonical projections onto the factors" and we say the maps fj "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).