`2021-04-09 15:00:00``2021-04-09 16:00:00``The OSU-OU Ring Theory Seminar``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.``Zoom (email cosmin@math.osu.edu for link)``OSU ASC Drupal 8``ascwebservices@osu.edu``America/New_York``public`

`2021-04-09 15:00:00``2021-04-09 16:00:00``The OSU-OU Ring Theory Seminar``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.``Zoom (email cosmin@math.osu.edu for link)``Department of Mathematics``math@osu.edu``America/New_York``public`**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.