# Irvin Cohen

**Irvin Sol Cohen** (1917 – February 14, 1955) was an American mathematician at the Massachusetts Institute of Technology who worked on local rings. He was a student of Oscar Zariski at Johns Hopkins University.

Irvin S. Cohen | |
---|---|

Born | 1917 |

Died | 14 February 1955 |

Nationality | |

Alma mater | Johns Hopkins University (Ph.D., 1942) |

Known for | Cohen-Macaulay rings, Cohen structure theorem, Cohen-Seidenberg theorems, unmixedness theorem, Cohen rings |

Scientific career | |

Fields | Mathematics |

Institutions | Massachusetts Institute of Technology |

Doctoral advisor | Oscar Zariski |

Doctoral students | R. Duncan Luce |

In his thesis he proved the Cohen structure theorem for complete Noetherian local rings.[1] In 1946 he proved the unmixedness theorem for power series rings. As a result, Cohen–Macaulay rings are named after him and Francis Sowerby Macaulay.

Cohen and Abraham Seidenberg published their Cohen–Seidenberg theorems, also known as the going-up and going-down theorems. He also coauthored articles with Irving Kaplansky. One of his doctoral students was R. Duncan Luce.

## Death

Cohen died unexpectedly in 1955 one week after having visited Zariski in Cambridge, apparently from suicide. Many years later Zariski said of his death:[1]

Many things are necessary to make a good scientist, a creative man, and left on his own Cohen found himself unproductive. Highly critical of himself and others, he believed that nothing he ever wrote was as good as his thesis. He became increasingly involved with abstract algebra until he found himself at a certain point without ground under his feet. He became disappointed in his work, and finally, fatally, in his own ability.

## Publications

- Cohen, I. S. (1946). "On the structure and ideal theory of complete local rings".
*Transactions of the American Mathematical Society*.**59**(1): 54–106. doi:10.2307/1990313. - Cohen, I.S.; Seidenberg, A. (1946). "Prime ideals and integral dependence".
*Bulletin of the American Mathematical Society*.**52**(4): 252–261. doi:10.1090/s0002-9904-1946-08552-3. MR 0015379. - Cohen, I.S.; Kaplansky, Irving (1946). "Rings With a Finite Number of Primes. I".
*Transactions of the American Mathematical Society*.**60**(3): 468–477. doi:10.2307/1990350. - Cohen, I.S. (1950). "Commutative rings with restricted minimum condition".
*Duke Mathematical Journal*.**71**(1). - Cohen, I.S. (1954). "Length of prime ideal chains".
*American Journal of Mathematics*.**76**(3): 654–668. doi:10.2307/2372708. - Cohen, I.S.; Kaplansky, Irving (1951). "Rings for which every module is a direct sum of cyclic modules".
*Mathematische Zeitschrift*.**54**(1): 97–101. - Cohen, I.S.; Zariski, Oscar (1957). "A fundamental inequality in the theory of extensions of valuations".
*Illinois Journal of Mathematics*.**1**(1): 1–8. doi:10.1215/ijm/1255378500.

## References

- Irvin Cohen at the Mathematics Genealogy Project
- Parikh, Carol (2014).
*The Unreal Life of Oscar Zariski*. Academic Press. p. 64. ASIN B01DUEBQSC. - "News and Notices".
*American Mathematical Monthly*.**62**(4): 296. 1955. JSTOR 2306718. - The Mathematical Association of America. "Report of the Treasurer for the Year 1954": 296. doi:10.1080/00029890.1955.11988629. Cite journal requires
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