1981/01/01 by Patrick F. Smith · 2 citations
Mathematics · #Rings, Modules, and Algebras #Commutative Algebra and Its Applications #Algebraic structures and combinatorial models #Mathematics #Injective function #Noetherian ring #Injective module #Homomorphism #Horizontal line test #Krull dimension #Noetherian #Prime ideal #Commutative ring #Pure mathematics #Associated prime #Commutative property #Prime (order theory) #Divisible group #Ideal (ethics) #Discrete mathematics #Radical of a ring #Ring (chemistry) #Principal ideal ring #Combinatorics #Algebra over a field #Geometry
paper · doi:10.1080/00927878108822627
openalex publication_date 1981/01/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
Let be a non-empty collection of right ideals of a ring R . A right R-module X is called -injective provided each R-homomorphism ϕ: A → X with A in can be lifted to an R-homomorphism θ: R→X . If R is a commutative Noetherian ring and . = Spec R then every -injective R-module is injective. On the other hand, if R is a commutative Noetherian integral domain of finite global dimension and a non-empty collection of prime ideals of R containing the zero ideal such that every -injective R-module is injective then = Spec R.