Speaker: Engin Buyukasik (Izmir Institute of Technology)
Abstract: A right module M is called direct-injective (or C2) if every submodule isomorphic to a summand of M is itself a summand. Dually, a module M is called direct-projective (or D2) if for every submodule N of M with M/N isomorphic to summand of M, then N is a summand of M. Recently, in a series of papers “simple” versions of the aforementioned modules have been investigated ([3], [4], [5]). These modules are termed as “simple-direct-injective” and “simple-direct-projective,” respectively.
In this talk, we shall discuss the structure of these modules over certain rings including the ring of integers. Besides, it will be shown that the rings whose simple-direct-injective right modules are simple-direct-projective are exactly the left perfect right H-rings, and that, for a commutative Noetherian ring, simple-direct-projective modules are simple-direct-injective if and only if simple-direct-injective modules are simple-direct-projective if and only if the ring is Artinian. These results are recently appeared in [1].
References
[1] B¨uy¨uka¸sık, E., Demir, O., Diril, M. (2021). On simple-direct modules. ¨ Comm. Algebra 49:864-876.
[2] Camillo, V. (1978). Homological independence of injective hulls of simple modules over commutative rings, Comm. Algebra 6:1459-1469.
[3] Camillo, V., Ibrahim, Y., Yousif, M., Zhou, Y. (2014). Simple-direct-injective modules, J. Algebra 420:39-53.
[4] Ibrahim, Y., Ko¸san, M. T., Quynh, T. C., Yousif, M. (2016). Simple-Direct-Projective Modules. Comm. Algebra 44:5163-5178.
[5] Ibrahim, Y., Ko¸san, M. T., Quynh, T. C., Yousif, M. (2017). Simple-direct-modules. Comm. Algebra 45:3643-3652.
