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The notion of pure subgroups were first investigated by Prufer in [9]. Purity has utmost importance in abelian group theory because it makes possible to use the methods of relative homological algebra as there are enough pure-injective and enough pure-projective groups. The purity concept was extended to modules over arbitrary rings by Cohn [3], Bourbaki [1], Butler and Horrocks [2] and Walker [11]. Stenström generalized the notion of purity to an abelian category with a (projective) generator in [10]. The notions of Rickart and dual Rickart were introduced and studied for modules by Lee, Rizvi and Roman [6, 7, 8]. Rickart and dual Rickart modules have been generalized to abelian categories by Crivei, Kör and Olteanu [4, 5]. In this work, (dual) purely Rickart objects are introduced as generalizations of (dual) Rickart objects in Gröthendieck categories. Examples showing the relations between (dual) relative Rickart objects and (dual) relative purely Rickart objects are given. It is shown that in a spectral category (dual) relative purely Rickart objects coincide with (dual) relative Rickart objects. (Co)products of (dual) relative purely Rickart objects are studied. Classes all of whose objects are (dual) relative purely Rickart are identified. Applications to comodule categories are given.
References
[1] N. Bourbaki, Elements of Mathematics, Commutative Algebra, Addison-Wesley Publishing Company, Advanced Book Program, Reading Massachusetts, 1972. Originally published as: Elements De Mathematique, Algebre Commutative, Hermann, Paris, 1969.
[2] M. C. R. Butler and G. Horrocks, Classes of Extensions and Resolutions, Phil. Trans. Royal Soc. of London, Series A 254 (1961), 155–222.
[3] P. M. Cohn, On the Free Product of Associative Rings, Mathematische Zeitschrift 71 (1959), 380–398.
[4] S. Crivei and A. Kör, Rickart and Dual Rickart Objects in Abelian Categories, Appl Categor Struct 24 (2016), 797–824.
[5] S. Crivei and G. Olteanu, Rickart and Dual Rickart Objects in Abelian Categories: Transfer via Functors, Appl Categor Struct 26 (2018), 681–698. \newpage
[6] G. Lee, S. T. Rizvi and C. S. Roman, Rickart Modules, Comm. Algebra 38 (2010), 4005–4027.  
[7] G. Lee, S. T. Rizvi and C. S. Roman, Dual Rickart Modules, Comm. Algebra 39 (2011), 4036–4058.  
[8] G. Lee, S. T. Rizvi and C. S. Roman, Direct Sums of Rickart Modules, J. Algebra 353 (2012), 62–78.  
[9] H. Prüfer, Untersuchungen  über die Zerlegbarkeit der abzaehlbaren prim aeren Abelschen Gruppen, Mathematische Zeitschrift 17 (1923), 35–61.
[10] B. T. Stenström, Pure submodules, Arkiv För Matematik 7 (10) (1966), 159–171.
[11] C. L. Walker, Relative Homological Algebra and Abelian Groups, Illinois J. Math. 10 (1966), 186–209