Journal of Linear and Topological Algebra Vol. 05, No. 02, 2016, 105- 109 On baer type criterion for C -dense, C -closed and quasi injectivity H. Barzegar a* , H. Arianpoor a a Department of Mathematics, Tafresh University, Tafresh 3951879611, Iran. Received 13 February 2016; Revised 21 May 2016; Accepted 10 September 2016. Abstract. For the subclasses M 1 and M 2 of monomorphisms in a concrete category C, if M 2 ⊆M 1 , then M 1 -injectivity implies 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. c ⃝ 2016 IAUCTB. All rights reserved. Keywords: C-Dense injective, C-closed injective, quasi injective. 2010 AMS Subject Classification: 08B30, 18G05. 1. Introduction The general theory of algebras and categories borrow techniques, ideas, and inspiration from older and more specialized branches of mathematics such as groups, rings, and modules. In this direction, Injectivity is one of the most central and important con- cepts which category theory inherited from homological and commutative algebra. The behaviour of this notion plays a central role in categorical model theory, notably in the characterization theorem for accessible categories with products, as the small-injectivity classes of locally presentable categories [1]. However, the study of injectivity with respect to different classes of monomorphisms is crucial in almost all categories. Throughout this paper C will denote a given concrete category (in which the objects are sets endowed with some additional structures and the morphisms are structure-preserving mappings) containing a zero object. The reader is refereed to [2] and [6] for some required * Corresponding author. E-mail address: h56bar@tafreshu.ac.ir (H. Barzegar). Print ISSN: 2252-0201 c ⃝ 2016 IAUCTB. All rights reserved. Online ISSN: 2345-5934 http://jlta.iauctb.ac.ir