mathematics

Irvin Cohen

Irvin Cohen is a mathematics topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Irvin Cohen rather than just read about it. In short: 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.

Key takeaways

Irvin Cohen belongs to mathematics; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect Irvin Cohen to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of Irvin Cohen from memory before moving on to harder problems.

Reference excerpt

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. In his thesis he proved the Cohen structure theorem for complete Noetherian local rings. 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. In 1950, he proved that a commutative ring whose prime ideals are all finitely generated is a Noetherian ring. 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:

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. JSTOR 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. JSTOR 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. JSTOR 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. doi:10.1007/BF01179851. S2CID 123538023. 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 (1955). "Report of the Treasurer for the Year 1954". The American Mathematical Monthly. 62 (4): 296. doi:10.1080/00029890.1955.11988629.

Worked examples

Example 1 — a first encounter with Irvin Cohen

Start with the simplest possible case. Write down what Irvin Cohen claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In mathematics, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Irvin Cohen before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Irvin Cohen ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Irvin Cohen

In research: Irvin Cohen appears in mathematics research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses Irvin Cohen in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: Irvin Cohen is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1917 births, 1955 deaths, 1955 suicides, so understanding it makes those chapters shorter.
In everyday life: Look for Irvin Cohen outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

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How to study Irvin Cohen in 20 minutes

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Frequently asked questions

What is Irvin Cohen in simple terms?

Why does Irvin Cohen matter?

Because it connects several mathematics ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Irvin Cohen?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Irvin Cohen.