On Baer type criterion for $C$-dense‎, ‎$C$-closed and quasi injectivity

محورهای موضوعی : History and biography

H. Barzegar ¹ , H. Arianpoor ²

1 - Department of Mathematics, Tafresh University, Tafresh 3951879611, Iran
2 - Department of Mathematics, Tafresh University, Tafresh 3951879611, Iran

تاریخ دریافت : 1394/11/24 تاریخ پذیرش : 1395/06/20 تاریخ انتشار : 1395/06/24

کلید واژه: $C$-dense injective&lrm, , &lrm, $C$-closed injective&lrm, , &lrm, quasi-injective,

چکیده مقاله :

‎For the subclasses $\mathcal{M}_1$ and $\mathcal{M}_2$ of‎‎monomorphisms in a concrete category $\mathcal{C}$‎, ‎if $\mathcal‎{M}_2\subseteq \mathcal{M}_1$‎, ‎then $\mathcal{M}_1$-injectivity‎‎implies $\mathcal{M}_2$-injectivity‎. ‎The Baer type criterion is about‎ ‎the converse of this fact‎. ‎In this paper‎, ‎we apply injectivity to the classes of $C$-dense‎, ‎$C$-closed‎ ‎monomorphisms‎. ‎The concept of quasi injectivity is also introduced here to‎ ‎investigte the Baer type criterion for these notions‎.

منابع و مأخذ:

[1] J. Adamek, and J. Rosicky, Locally Presentable and Accessible Categories. Cambridge University Press, Cambridge, 1994.

[2] J. Adamek, H. Herrlich and G.E. Strecker, Abstract and Concrete Categories. John Wiley and Sons, Inc., 1990.

[3] J. Ahsan, Monoids characterized by their quasi-injective S-systems. Semigroup Forum 36 (3) (1987), 285-292.

[4] H. Barzegar, M.M. Ebrahimi and M. Mahmoudi, Essentiality and Injectivity. Appl. Categor. Struct. 18 (1) (2010), 73-83.

[5] P. Berthiaume, The injective envelope of S-Sets. Canad. Math. Bull. 10 (2) (1967), 261-273.

[6] T.S. Blyth, Categories. Essex CM202JE, England, 1986.

[7] D. Dikranjan and W. Tholen, Categorical structure of closure operators, with applications to topology, algebra, and discrete mathematics. Mathematics and Its Applications, Kluwer Academic Publ., 1995.

[8] W. Tholen, Injective objects and cogenerating sets. J. Algebra, 73 (1) (1981), 139-155.