User:Barto

Please, post any opinions about the below on my talk page. There are some choices to be made, and I don't feel authorized to make them without anyone's consent.

Modules

 * Organize Direct Product of Modules is Module, Finite Direct Product of Modules is Module, Module on Cartesian Product, Vector Space of All Mappings
 * Consider renaming Definition:Free Module Indexed by Set to something less pedantic (name is used in some other articles; better decide quickly before there are more)

Direct Product and Direct Sum
and the naming of those subdefinitions should be consistent: Finite Case, General Case or something like that.
 * Make sure that the definition pages for direct products include a definition for arbitrary families. Ideally, there should be:
 * A definition for two components
 * A definition for a finite number of components
 * A definition for general families
 * (Optionally: a definition for countable families)


 * Be consistent in naming pages:
 * product vs. sum: product is the primary notion, sum is derived from it. It is thus impossible to define the direct sum without defining the direct product; unless when they coincide and allow for ambiguous naming.
 * External vs. internal: If nothing is specified, external has to be understood. IMO external can be removed from titles; it's unnecessarily pedantic; though I could live with it if it stays.
 * Word order: Direct Sum of X's vs. X Direct Sum?
 * The following table shows the jolly inconsistencies (no, I didn't misplace the entries):

Please, feel free to add missing entries. These are the only definitions I found so far (not including transclusions). Some definitions are still to be extracted from a theorem that's being refactored.

Support

 * Elements with Support in Ideal form Submagma of Direct Product: fix notion of ideal and link
 * Elements of Finite Support form Submagma of Direct Product same

Polynomials

 * Clean up Definition:Polynomial (Abstract Algebra)
 * Add a definition using Definition:Monoid Ring

Modules

 * Define modules and unitary modules using Definition:Endomorphism Ring of Abelian Group, see Morphism to Endomorphism Ring Defines Module
 * Determining Definition:Multilinear Mapping using basises, use this in Definition:Monoid Ring
 * Correspondence between Definition:Multilinear Mapping and linear maps to Definition:Module of Multilinear Mappings.

Definition-level

 * Free Module Indexed by Set is Free
 * Canonical Basis of Free Module Indexed by Set is Basis
 * Universal Property of Free Modules
 * Universal Property of Direct Product of Modules
 * Universal Property of Direct Sum of Modules
 * Endomorphism Ring of Abelian Group is Ring

Spaces of Morphisms

 * Definition:Group of Homomorphisms between Abelian Groups
 * Definition:Endomorphism Ring of Abelian Group
 * Definition:Module of Homomorphisms between Modules
 * Definition:Module of Multilinear Mappings
 * Definition:Endomorphism Ring of Module
 * Note that there is already a general Definition:Group of Automorphisms for algebraic structures with one operation.