if "b" is a ring without an identity element, we can extend the definition of k as follows. let "a" = "b"⊕z be the extension of "b" to a ring with unity obtaining by adjoining an identity element (0,1). there is a short exact sequence "b" → "a" → z and we define k("b") to be the kernel of the corresponding map k("a") → k(z) = z.