To every right or left Bol loop corresponds a middle Bol loop. In this paper, the cores of right Bol loops (RBL) and its corresponding middle Bol loops (MBL) were studied. Their algebraic connections were considered. It was shown that the core of a RBL is elastic and right idempotent. The core of a RBL was found to be alternative (or left idempotent) if and only if its corresponding MBL is right symmetric. If a MBL is right (left) symmetric, then, the core of its corresponding RBL is a medial (semimedial). The core of a middle Bol loop has the left inverse property (automorphic inverse property, right idempotence resp.) if and only if its corresponding RBL has the super anti-automorphic inverse property (automorphic inverse property, exponent 2 resp.). If a RBL is of exponent 2, then, the core of its corresponding MBL is left idempotent. If a RBL is of exponent 2 then: the core of a MBL has the left alternative property (right alternative property) if and only if its corresponding RBL has the cross inverse property (middle symmetry). Some other similar results were derived for RBL of exponent 3.