M"/Im ~ Since @* @ N = M"/Im $ and let the canonical homomorphlsm. is monlc, that ~*~* = O; have ~*~*(p) = ~@l M let o: M - - > M / I m N = M/Im ~ is epic, let o = O. , @*(o) = ~o = O. M, we observe N = M , p = 1M , we Im ~ C Ker 9- Conversely, be the canonical homomorphism, and put in (ii). , >M/Is / / / OT / M" is a commutative diagram. sequence (1) is exact. 7 Theorem: Let Hence Ker @ C # NI,N,N" • R M.

# Advanced algebra by Rotman J.J.

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