Correct classes of modules

• Опубліковано в: : Algebra and Discrete Mathematics
• Дата: 2004
• Автор: Wisbauer, R.
• Формат: Стаття
• Мова: Англійська
• Опубліковано: Інститут прикладної математики і механіки НАН України 2004
• Онлайн доступ: https://nasplib.isofts.kiev.ua/handle/123456789/156603
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Цитувати: Correct classes of modules / R. Wisbauer // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 4. — С. 106–118. — Бібліогр.: 18 назв. — англ.

For a ring R, call a class C of R-modules (pure-) mono-correct if for any M, N ∈ C the existence of (pure) monomorphisms M → N and N → M implies M ≃ N. Extending results and ideas of Rososhek from rings to modules, it is shown that, for an R-module M, the class σ[M] of all M-subgenerated modules is mono-correct if and only if M is semisimple, and the class of all weakly M-injective modules is mono-correct if and only if M is locally noetherian. Applying this to the functor ring of R-Mod provides a new proof that R is left pure semisimple if and only if R-Mod is pure-mono-correct. Furthermore, the class of pure-injective Rmodules is always pure-mono-correct, and it is mono-correct if and only if R is von Neumann regular. The dual notion epi-correctness is also considered and it is shown that a ring R is left perfect if and only if the class of all flat R-modules is epi-correct. At the end some open problems are stated.