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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. An ordered ring, also called a totally-ordered ring, is a partially-ordered ring (A, leq) where le is additionally a total order. Every ordered ring is an f-ring, so every subdirect union of ordered rings is also an f-ring. Assuming the axiom of choice, a theorem of Birkhoff shows the converse, and that an l-ring is an f-ring if and only if it is l-isomorphic to a subdirect union of ordered rings. Some mathematicians take this to be the definition of an f-ring.