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The concept of right Rickart rings (or right p.p. rings) has been extensively studied in the literature. In this article, we study the notion of Rickart modules in the general module theoretic setting by utilizing the endomorphism ring of a module. We provide several characterizations of Rickart modules and study their properties. It is shown that the class of rings R for which every right R-module is Rickart is precisely that of semisimple artinian rings, while the class of rings R for which every free R-module is Rickart is precisely that of right hereditary rings. Connections between a Rickart module and its endomorphism ring are studied. A characterization of precisely when the endomorphism ring of a Rickart module will be a right Rickart ring is provided. We prove that a Rickart module with no infinite set of nonzero orthogonal idempotents in its endomorphism ring is precisely a Baer module. We show that a finitely generated module over a principal ideal domain (PID) is Rickart exactly if it is either semisimple or torsion-free. Examples which delineate the concepts and results are provided.

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A ring R is called Armendariz if whenever the product of any two polynomials in R[x] over R is zero, then so is the product of any pair of coefficients from the two polynomials. Such rings have been extensively studied in literature. For a ring endomorphism α, we introduce the notion of α-Armendariz rings by considering the polynomials in the skew polynomial ring R[x; α] in place of the ring R[x]. A number of properties of this generalization are established, and connections of properties of an α-Armendariz ring R with those of the ring R[x; α] are investigated. In particular, among other results, we show that there is a strong connection of the Baer property and the p.p.-property (principal ideals are projective) of the two rings, respectively. Several known results follow as consequences of our results.