The theorem was first published by Thoralf Skolem in 1927 in his paper Zur Theorie der assoziativen Zahlensysteme (German: On the theory of associative number systems) and later rediscovered by Emmy Noether.
In a general formulation, let A and B be simple rings, and let k be the centre of B. Notice that k is a field since given x nonzero in k, the simplicity of B implies that the nonzero two-sided ideal Bx is the whole of B, and hence that x is a unit. Suppose further that the dimension of B over k is finite, i.e. that B is a central simple algebra. Then given k-algebra homomorphisms
- f, g : A → B
- g(a) = b · f(a) · b−1.
First suppose . Then f and g define the actions of A on ; let denote the A-modules thus obtained. Any two simple A-modules are isomorphic and are finite direct sums of simple A-modules. Since they have the same dimension, it follows that there is an isomorphism of A-modules . But such b must be an element of . For the general case, note that is a matrix algebra and thus by the first part this algebra has an element b such that
for all and . Taking , we find
for all z. That is to say, b is in and so we can write . Taking this time we find
which is what was sought.
- Lorenz (2008) p.173
- Farb, Benson; Dennis, R. Keith (1993). Noncommutative Algebra. Springer. ISBN 9780387940571.
- Gille & Szamuely (2006) p.40
- Lorenz (2008) p.174
- Skolem, Thoralf (1927). "Zur Theorie der assoziativen Zahlensysteme". Skrifter Oslo (in German) (12): 50. JFM 54.0154.02.
- A discussion in Chapter IV of Milne, class field theory 
- Gille, Philippe; Szamuely, Tamás (2006). Central simple algebras and Galois cohomology. Cambridge Studies in Advanced Mathematics 101. Cambridge: Cambridge University Press. ISBN 0-521-86103-9. Zbl 1137.12001.
- Lorenz, Falko (2008). Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics. Springer. ISBN 978-0-387-72487-4. Zbl 1130.12001.