This work is based on the results and methods of the author's earlier study of Jordan \(A\)-algebras [Algebra Logika 26, No. 6, 731--755 (1987; Zbl 0648.17008)]. The term algebra is used when working over a commutative associative ring with \(1/2\), while the term ring is used when working over the integers. If \(J\) is an algebra, \(S\) is a subalgebra of \(J\), and \(I\) is an ideal of \(S\), then \(S/I\) is called a factor of the algebra \(J\). Singled out from the results in this work is the following basic one: If some factor of an arbitrary Jordan ring contains an \(A\)-subalgebra, then the ring itself contains an \(A\)-subalgebra, the center of which lies in the center of the whole ring. This result is used to study free Jordan algebras and also representations of Jordan algebras. In particular, the author proves: (1) A free Jordan ring from more than two generators contains an \(A\)-subalgebra, the center of which lies in the center of the whole ring. (2) A free Jordan algebra from more than two generators is not prime and has a nonzero center.