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Using the Bockstein spectral sequence developed previously by the authors, we compute the ring    E  R    (  n  )   *    (  B  O   (  q  )   )explicitly. We then use this calculation to show that the ring spectrum    M  O  [   2   n  +  1    ]    is    E  R  (  n  )   ‐orientable (but not    E  R  (  n  +  1  )   ‐orientable), where    M  O  [   2   n  +  1    ]    is defined as the Thom spectrum for the self‐map of    B  O    given by multiplication by    2   n  +  1    .

The ER ( n )‐cohomology of BO ( q ) and real Johnson–Wilson orientations for vector bundles