Abelian Categories with Applications to Rings and Modules |
Common terms and phrases
5-category A-module a₁ abelian category abelian groups additive functor Ann(x artinian assertions are equivalent assume b₁ canonical functor canonical inclusion canonical morphism Chapter cogenerator coker cokernel commutative diagram commutative ring contains COROLLARY covariant functor deduce defined denote direct product direct sum direct summand element epimorphism essential extension exact sequence Exercise F-closed finite type following assertions full and faithful full subcategory functorial morphism Gabriel 59 Grothendieck category Hence inductive limit injective envelope injective object isomorphism ker f ker ƒ kernel left ideal left prime Let F localizing subcategory localizing system Math maximal module monomorphism Moreover morphism f natural number non-zero subobject Obviously P₁ phism Popescu preadditive category prime ideal projective Proof PROPOSITION prove quotient object respectively right adjoint ring morphism s.n.-category Section semisimple set of subobjects simple object STC(A structural morphisms submodule subset Theorem topology u₁ unique morphism X₁ Y₁