2025/02/27 by Zhang, Xiaolei
#FOS: Mathematics #Rings and Algebras (math.RA)
paper · doi:10.48550/arxiv.2502.19641
We show that the condition of being categorical in a tail of cardinals can be characterized for the class of R-modules of the form \Add(M). More precisely, let R be a ring and M be an R-module which can be generated by ≤ ℵ elements. Then \Add(M) is κ-categorical in all κ>\Vert R\Vert+ℵ+ℵ0 if and only if \Add(M) is κ-categorical in some κ>\Vert R\Vert+ℵ+ℵ0; if and only if every R-module of cardinal κ in \Add(M) is M-free for all κ>\Vert R\Vert+ℵ+ℵ0; if and only if every R-module of cardinal (\Vert R\Vert+ℵ+ℵ0)+ in \Add(M) is M-free. As an application, we show that the class of pure-projective R-modules is categorical in some (all) big cardinal if and only if R is R≅ Mn(D) where D is a division ring and n≥ 1; the class of semisimple R-modules is categorical in some (all) big cardinal if and only if R admits a unique simple module up to isomorphism, partly answering a question proposed in [5, Mazari-Armida M., Characterizing categoricity in several classes of modules. J. Algebra 617, 382-401 (2023)].