Ring Theory Seminar

Christopher Schafhauser, Department of Pure Mathematics, University of Waterloo

“Goldie’s Theorem on semiprime rings”

Recall that R is a right Ore ring if its set S = C(R) of cancelable elements satisfies the right Ore condition. In this case we can form the right classical ring of fractions Q = RS−1, and there is a natural inclusion R  Q such that S ⊆ Q× and each element of Q has the form as−1,forsomea∈Rands∈S. WesawthatifRisadomainthenQisadivisionring,but this fails if R is not right Ore — in fact, in the non-Ore case we can’t form the right ring of fractions. But we can still ask whether or not R embeds in some ring Q such that S ⊆ Q× and each element of Q has the form as−1. In this case R is called a right order in Q.

Our aim now is to determine when R is a right order in a semisimple ring. For this, there are necessary and sufficient conditions given by Goldie’s Theorem.

MC 5403