mathematics

Neal Henry McCoy

Neal Henry McCoy 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 Neal Henry McCoy rather than just read about it. In short: Neal Henry McCoy (March 6, 1905 – January 5, 2001) was an American mathematician, university professor, and author of several textbooks for mathematics undergraduates. His 1960 textbook Introduction to Modern Algebra has gone through several editions and has been translated into many foreign languages.

Key takeaways

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Reference excerpt

Neal Henry McCoy (March 6, 1905 – January 5, 2001) was an American mathematician, university professor, and author of several textbooks for mathematics undergraduates. His 1960 textbook Introduction to Modern Algebra has gone through several editions and has been translated into many foreign languages.

Early life and education McCoy was born into a homesteading family in Waukomis, Oklahoma (which was part of the Oklahoma Territory from 1890 to 1907). He had an older sister Dorothy McCoy (1903–2001), who became the first woman to receive a Ph.D. in mathematics from the University of Iowa. As an infant, he moved in 1906 with his mother and sister to Chesapeake, Missouri after the death of his father. After graduating with a bachelor's degree from Baylor University, he became a graduate student at the University of Iowa. There he graduated in 1929 with a Ph.D. in mathematics with adviser Edward Wilson Chittenden. McCoy's Ph.D. thesis On commutation formulas in the algebra of quantum mechanics was published in abbreviated form in the Transactions of the American Mathematical Society.

Career For two academic years from 1929 to 1931, McCoy studied at Princeton University as a National Research Fellow. He joined in 1931 the staff of the mathematics department of Northampton's Smith College, There he was promoted to full professor in 1942 and was appointed to the Gates Chair in Mathematics in 1963, retiring as professor emeritus in 1970. McCoy did research on abstract algebra, especially the theory of rings and matrices with elements in rings, and algebraic aspects of quantum mechanics. From 1951 to 1954 he was the editor-in-chief of the Duke Mathematical Journal. He was an associate editor for the American Mathematical Monthly and the Pi Mu Epsilon Journal.

McCoy ring In 1942, McCoy proved the following theorem:Let R be a commutative ring with multiplicative unit, and let f(x) be a zero divisor in the polynomial ring R[x]. Then there is a nonzero element r ∈ R with f(x)r = 0. In 1997 Mangesh B. Rege and Sima Chhawchharia introduced definitions of rings, which in 2006, independently and motivated by McCoy's 1942 theorem, were re-introduced by Pace P. Nielsen, who gave the concepts the names "right McCoy ring", "left McCoy ring", and "McCoy ring" (a ring which is both right McCoy and left McCoy).

Rings and Ideals (1948) McCoy wrote the 8th book in the series of Carus Mathematical Monographs published by the Mathematical Association of America. The book is almost entirely self-contained. In a 1948 review, André Weil praised the book for the optimized simplicity of the mathematical proofs, an "easy and readable style", and "skillful use of examples". In a 1949 review, C. C. MacDuffee praised the book — "as an introduction to the powerful and highly abstract method of thinking which now characterizes modern algebra, it is a gem."

Personal life In 1929 in Iowa, Neal H. McCoy married Ardis Hollingsworth (1904–1988). Their only son, Paul Albert McCoy, was killed in 1957 at age 22 in an automobile accident.

Selected publications

Articles McCoy, Neal H. (1929). "On Commutation Rules in the Algebra of Quantum Mechanics". Proceedings of the National Academy of Sciences of the United States of America. 15 (3): 200–202. Bibcode:1929PNAS...15..200M. doi:10.1073/pnas.15.3.200. JSTOR 85232. PMC 522434. PMID 16577167. McCoy, Neal H. (1932). "On the Function in Quantum Mechanics Which Corresponds to a Given Function in Classical Mechanics". Proceedings of the National Academy of Sciences. 18 (11): 674–676. Bibcode:1932PNAS...18..674M. doi:10.1073/pnas.18.11.674. PMC 1076309. PMID 16577495. McCoy, Neal H. (1934). "On quasi-commutative matrices". Transactions of the American Mathematical Society. 36 (2): 327–340. doi:10.1090/s0002-9947-1934-1501746-8. McCoy, N. H. (1936). "On the characteristic roots of matric polynomials" (PDF). Bulletin of the American Mathematical Society. 42 (8): 592–600. doi:10.1090/s0002-9904-1936-06372-X. (See characteristic root and matrix polynomial.) McCoy, N. H.; Montgomery, Deane (1937). "A representation of generalized Boolean rings". Duke Mathematical Journal. 3 (3). doi:10.1215/S0012-7094-37-00335-1. McCoy, Neal H. (1938). "Subrings of infinite direct sums". Duke Mathematical Journal. 4 (3). doi:10.1215/S0012-7094-38-00441-7. McCoy, Neal H. (1939). "A theorem on matrices over a commutative ring". Bulletin of the American Mathematical Society. 45 (10): 740–744. doi:10.1090/s0002-9904-1939-07070-5. McCoy, Neal H. (1939). "Concerning matrices with elements in a commutative ring". Bulletin of the American Mathematical Society. 45 (4): 280–284. doi:10.1090/s0002-9904-1939-06957-7. McCoy, N. H. (1939). "Generalized regular rings" (PDF). Bulletin of the American Mathematical Society. 45 (2): 175–178. doi:10.1090/s0002-9904-1939-06933-4. McCoy, N. H. (1942). "Remarks on Divisors of Zero". The American Mathematical Monthly. 49 (5): 286–295. doi:10.1080/00029890.1942.11991226. McCoy, Neal H. (1945). "Subdirectly irreducible commutative rings". Duke Mathematical Journal. 12 (2). doi:10.1215/S0012-7094-45-01232-4. Brown, Bailey; McCoy, Neal H. (1946). "Rings with unit element which contain a given ring". Duke Mathematical Journal. 13. doi:10.1215/s0012-7094-46-01302-6. Brown, Bailey; McCoy, Neal H. (1947). "Radicals and Subdirect Sums". American Journal of Mathematics. 69 (1): 46–58. doi:10.2307/2371653. JSTOR 2371653. McCoy, Neal H. (1947). "Subdirect sums of rings". Bulletin of the American Mathematical Society. 53 (9): 856–877. doi:10.1090/s0002-9904-1947-08867-4. McCoy, Neal H. (1949). "Prime Ideals in General Rings". American Journal of Mathematics. 71 (4): 823–833. doi:10.2307/2372366. JSTOR 2372366. Brown, Bailey; McCoy, Neal H. (1950). "Some theorems on groups with applications to ring theory". Transactions of the American Mathematical Society. 69: 302–311. doi:10.1090/S0002-9947-1950-0038952-7. McCoy, Neal H. (1957). "A note on finite unions of ideals and subgroups". Proceedings of the American Mathematical Society. 8 (4): 633–637. doi:10.1090/s0002-9939-1957-0086803-9. Brown, Bailey; McCoy, Neal H. (1958). "Prime ideals in nonassociative rings". Transactions of the American Mathematical Society. 89: 245–255. doi:10.1090/s0002-9947-1958-0096713-4.

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Worked examples

Example 1 — a first encounter with Neal Henry McCoy

Start with the simplest possible case. Write down what Neal Henry McCoy 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 Neal Henry McCoy 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 Neal Henry McCoy 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 Neal Henry McCoy

In research: Neal Henry McCoy 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 Neal Henry McCoy 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: Neal Henry McCoy is common in secondary-school and first-year university syllabi. It links to neighbouring topics 1905 births, 2001 deaths, 20th-century American mathematicians, so understanding it makes those chapters shorter.
In everyday life: Look for Neal Henry McCoy 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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Frequently asked questions

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Why does Neal Henry McCoy 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.

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