In mathematics, the quaternions are a number system that extends the complex numbers. They were first described by Irish mathematician William Rowan Hamilton in 1843 and applied to mechanics in three-dimensional space. A feature of quaternions is. Hamilton defined a quaternion as the quotient of two directed lines in a three-dimensional space or equivalently as the quotient of two vectors. Quaternions are represented in the form: a + b i + c j + d k where a, b, c, d are real numbers, i, j, k are the fundamental quaternion units. Quaternions find uses in both pure and applied mathematics, in particular for calculations involving three-dimensional rotations such as in three-dimensional computer graphics, computer vision, crystallographic texture analysis. In practical applications, they can be used alongside other methods, such as Euler angles and rotation matrices, or as an alternative to them, depending on the application. In modern mathematical language, quaternions form a four-dimensional associative normed division algebra over the real numbers, therefore a domain.
In fact, the quaternions were the first noncommutative division algebra. The algebra of quaternions is denoted by H, or in blackboard bold by H, it can be given by the Clifford algebra classifications Cℓ0,2 ≅ Cℓ03,0. The algebra ℍ holds a special place in analysis since, according to the Frobenius theorem, it is one of only two finite-dimensional division rings containing the real numbers as a proper subring, the other being the complex numbers; these rings are Euclidean Hurwitz algebras, of which quaternions are the largest associative algebra. Further extending the quaternions yields the non-associative octonions, the last normed division algebra over the reals; the unit quaternions can be thought of as a choice of a group structure on the 3-sphere S3 that gives the group Spin, isomorphic to SU and to the universal cover of SO. Quaternions were introduced by Hamilton in 1843. Important precursors to this work included Euler's four-square identity and Olinde Rodrigues' parameterization of general rotations by four parameters, but neither of these writers treated the four-parameter rotations as an algebra.
Carl Friedrich Gauss had discovered quaternions in 1819, but this work was not published until 1900. Hamilton knew that the complex numbers could be interpreted as points in a plane, he was looking for a way to do the same for points in three-dimensional space. Points in space can be represented by their coordinates, which are triples of numbers, for many years he had known how to add and subtract triples of numbers. However, Hamilton had been stuck on the problem of division for a long time, he could not figure out. The great breakthrough in quaternions came on Monday 16 October 1843 in Dublin, when Hamilton was on his way to the Royal Irish Academy where he was going to preside at a council meeting; as he walked along the towpath of the Royal Canal with his wife, the concepts behind quaternions were taking shape in his mind. When the answer dawned on him, Hamilton could not resist the urge to carve the formula for the quaternions, i 2 = j 2 = k 2 = i j k = − 1 into the stone of Brougham Bridge as he paused on it.
Although the carving has since faded away, there has been an annual pilgrimage since 1989 called the Hamilton Walk for scientists and mathematicians who walk from Dunsink Observatory to the Royal Canal bridge in remembrance of Hamilton's discovery. On the following day, Hamilton wrote a letter to his friend and fellow mathematician, John T. Graves, describing the train of thought that led to his discovery; this letter was published in a letter to a science magazine. An electric circuit seemed to close, a spark flashed forth. Hamilton called a quadruple with these rules of multiplication a quaternion, he devoted most of the remainder of his life to studying and teaching them. Hamilton's treatment is more geometric than the modern approach, which emphasizes quaternions' algebraic properties, he founded a school of "quaternionists", he tried to popularize quaternions in several books. The last and longest of his books, Elements of Quaternions, was 800 pages long. After Hamilton's death, his student Peter Tait continued promoting quaternions.
At this time, quaternions were a mandatory examination topic in Dublin. Topics in physics and geometry that would now be described using vectors, such as kinematics in space and Maxwell's equations, were described in terms of quaternions. There was a professional research association, the Quaternion Society, devoted to the study of quaternions and other hypercomplex number systems. From the mid-1880s, quaternions began to be displaced by vector analysis, developed by Josiah Willard Gibbs, Oliver Heaviside, Hermann von Helmholtz. Vector analys
Mathematics includes the study of such topics as quantity, structure and change. Mathematicians use patterns to formulate new conjectures; when mathematical structures are good models of real phenomena mathematical reasoning can provide insight or predictions about nature. Through the use of abstraction and logic, mathematics developed from counting, calculation and the systematic study of the shapes and motions of physical objects. Practical mathematics has been a human activity from as far back; the research required to solve mathematical problems can take years or centuries of sustained inquiry. Rigorous arguments first appeared in Greek mathematics, most notably in Euclid's Elements. Since the pioneering work of Giuseppe Peano, David Hilbert, others on axiomatic systems in the late 19th century, it has become customary to view mathematical research as establishing truth by rigorous deduction from appropriately chosen axioms and definitions. Mathematics developed at a slow pace until the Renaissance, when mathematical innovations interacting with new scientific discoveries led to a rapid increase in the rate of mathematical discovery that has continued to the present day.
Mathematics is essential in many fields, including natural science, medicine and the social sciences. Applied mathematics has led to new mathematical disciplines, such as statistics and game theory. Mathematicians engage in pure mathematics without having any application in mind, but practical applications for what began as pure mathematics are discovered later; the history of mathematics can be seen as an ever-increasing series of abstractions. The first abstraction, shared by many animals, was that of numbers: the realization that a collection of two apples and a collection of two oranges have something in common, namely quantity of their members; as evidenced by tallies found on bone, in addition to recognizing how to count physical objects, prehistoric peoples may have recognized how to count abstract quantities, like time – days, years. Evidence for more complex mathematics does not appear until around 3000 BC, when the Babylonians and Egyptians began using arithmetic and geometry for taxation and other financial calculations, for building and construction, for astronomy.
The most ancient mathematical texts from Mesopotamia and Egypt are from 2000–1800 BC. Many early texts mention Pythagorean triples and so, by inference, the Pythagorean theorem seems to be the most ancient and widespread mathematical development after basic arithmetic and geometry, it is in Babylonian mathematics that elementary arithmetic first appear in the archaeological record. The Babylonians possessed a place-value system, used a sexagesimal numeral system, still in use today for measuring angles and time. Beginning in the 6th century BC with the Pythagoreans, the Ancient Greeks began a systematic study of mathematics as a subject in its own right with Greek mathematics. Around 300 BC, Euclid introduced the axiomatic method still used in mathematics today, consisting of definition, axiom and proof, his textbook Elements is considered the most successful and influential textbook of all time. The greatest mathematician of antiquity is held to be Archimedes of Syracuse, he developed formulas for calculating the surface area and volume of solids of revolution and used the method of exhaustion to calculate the area under the arc of a parabola with the summation of an infinite series, in a manner not too dissimilar from modern calculus.
Other notable achievements of Greek mathematics are conic sections, trigonometry (Hipparchus of Nicaea, the beginnings of algebra. The Hindu–Arabic numeral system and the rules for the use of its operations, in use throughout the world today, evolved over the course of the first millennium AD in India and were transmitted to the Western world via Islamic mathematics. Other notable developments of Indian mathematics include the modern definition of sine and cosine, an early form of infinite series. During the Golden Age of Islam during the 9th and 10th centuries, mathematics saw many important innovations building on Greek mathematics; the most notable achievement of Islamic mathematics was the development of algebra. Other notable achievements of the Islamic period are advances in spherical trigonometry and the addition of the decimal point to the Arabic numeral system. Many notable mathematicians from this period were Persian, such as Al-Khwarismi, Omar Khayyam and Sharaf al-Dīn al-Ṭūsī. During the early modern period, mathematics began to develop at an accelerating pace in Western Europe.
The development of calculus by Newton and Leibniz in the 17th century revolutionized mathematics. Leonhard Euler was the most notable mathematician of the 18th century, contributing numerous theorems and discoveries; the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, differential geometry, matrix theory, number theory, statistics. In the early 20th century, Kurt Gödel transformed mathematics by publishing his incompleteness theorems, which show that any axiomatic system, consistent will contain unprovable propositions. Mathematics has since been extended, there has been a fruitful interaction between mathematics and science, to
Cambridge University Press
Cambridge University Press is the publishing business of the University of Cambridge. Granted letters patent by King Henry VIII in 1534, it is the world's oldest publishing house and the second-largest university press in the world, it holds letters patent as the Queen's Printer. The press mission is "to further the University's mission by disseminating knowledge in the pursuit of education and research at the highest international levels of excellence". Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. With a global sales presence, publishing hubs, offices in more than 40 countries, it publishes over 50,000 titles by authors from over 100 countries, its publishing includes academic journals, reference works and English language teaching and learning publications. Cambridge University Press is a charitable enterprise that transfers part of its annual surplus back to the university. Cambridge University Press is both the oldest publishing house in the world and the oldest university press.
It originated from letters patent granted to the University of Cambridge by Henry VIII in 1534, has been producing books continuously since the first University Press book was printed. Cambridge is one of the two privileged presses. Authors published by Cambridge have included John Milton, William Harvey, Isaac Newton, Bertrand Russell, Stephen Hawking. University printing began in Cambridge when the first practising University Printer, Thomas Thomas, set up a printing house on the site of what became the Senate House lawn – a few yards from where the press's bookshop now stands. In those days, the Stationers' Company in London jealously guarded its monopoly of printing, which explains the delay between the date of the university's letters patent and the printing of the first book. In 1591, Thomas's successor, John Legate, printed the first Cambridge Bible, an octavo edition of the popular Geneva Bible; the London Stationers objected strenuously. The university's response was to point out the provision in its charter to print "all manner of books".
Thus began the press's tradition of publishing the Bible, a tradition that has endured for over four centuries, beginning with the Geneva Bible, continuing with the Authorized Version, the Revised Version, the New English Bible and the Revised English Bible. The restrictions and compromises forced upon Cambridge by the dispute with the London Stationers did not come to an end until the scholar Richard Bentley was given the power to set up a'new-style press' in 1696. In July 1697 the Duke of Somerset made a loan of £200 to the university "towards the printing house and presse" and James Halman, Registrary of the University, lent £100 for the same purpose, it was in Bentley's time, in 1698, that a body of senior scholars was appointed to be responsible to the university for the press's affairs. The Press Syndicate's publishing committee still meets and its role still includes the review and approval of the press's planned output. John Baskerville became University Printer in the mid-eighteenth century.
Baskerville's concern was the production of the finest possible books using his own type-design and printing techniques. Baskerville wrote, "The importance of the work demands all my attention. Caxton would have found nothing to surprise him if he had walked into the press's printing house in the eighteenth century: all the type was still being set by hand. A technological breakthrough was badly needed, it came when Lord Stanhope perfected the making of stereotype plates; this involved making a mould of the whole surface of a page of type and casting plates from that mould. The press was the first to use this technique, in 1805 produced the technically successful and much-reprinted Cambridge Stereotype Bible. By the 1850s the press was using steam-powered machine presses, employing two to three hundred people, occupying several buildings in the Silver Street and Mill Lane area, including the one that the press still occupies, the Pitt Building, built for the press and in honour of William Pitt the Younger.
Under the stewardship of C. J. Clay, University Printer from 1854 to 1882, the press increased the size and scale of its academic and educational publishing operation. An important factor in this increase was the inauguration of its list of schoolbooks. During Clay's administration, the press undertook a sizeable co-publishing venture with Oxford: the Revised Version of the Bible, begun in 1870 and completed in 1885, it was in this period as well that the Syndics of the press turned down what became the Oxford English Dictionary—a proposal for, brought to Cambridge by James Murray before he turned to Oxford. The appointment of R. T. Wright as Secretary of the Press Syndicate in 1892 marked the beginning of the press's development as a modern publishing business with a defined editorial policy and administrative structure, it was Wright who devised the plan for one of the most distinctive Cambridge contributions to publishing—the Cambridge Histories. The Cambridge Modern History was published
In mathematics, a group is a set equipped with a binary operation which combines any two elements to form a third element in such a way that four conditions called group axioms are satisfied, namely closure, associativity and invertibility. One of the most familiar examples of a group is the set of integers together with the addition operation, but groups are encountered in numerous areas within and outside mathematics, help focusing on essential structural aspects, by detaching them from the concrete nature of the subject of the study. Groups share a fundamental kinship with the notion of symmetry. For example, a symmetry group encodes symmetry features of a geometrical object: the group consists of the set of transformations that leave the object unchanged and the operation of combining two such transformations by performing one after the other. Lie groups are the symmetry groups used in the Standard Model of particle physics; the concept of a group arose from the study of polynomial equations, starting with Évariste Galois in the 1830s.
After contributions from other fields such as number theory and geometry, the group notion was generalized and established around 1870. Modern group theory—an active mathematical discipline—studies groups in their own right. To explore groups, mathematicians have devised various notions to break groups into smaller, better-understandable pieces, such as subgroups, quotient groups and simple groups. In addition to their abstract properties, group theorists study the different ways in which a group can be expressed concretely, both from a point of view of representation theory and of computational group theory. A theory has been developed for finite groups, which culminated with the classification of finite simple groups, completed in 2004. Since the mid-1980s, geometric group theory, which studies finitely generated groups as geometric objects, has become an active area in group theory; the modern concept of an abstract group developed out of several fields of mathematics. The original motivation for group theory was the quest for solutions of polynomial equations of degree higher than 4.
The 19th-century French mathematician Évariste Galois, extending prior work of Paolo Ruffini and Joseph-Louis Lagrange, gave a criterion for the solvability of a particular polynomial equation in terms of the symmetry group of its roots. The elements of such a Galois group correspond to certain permutations of the roots. At first, Galois' ideas were rejected by his contemporaries, published only posthumously. More general permutation groups were investigated in particular by Augustin Louis Cauchy. Arthur Cayley's On the theory of groups, as depending on the symbolic equation θn = 1 gives the first abstract definition of a finite group. Geometry was a second field in which groups were used systematically symmetry groups as part of Felix Klein's 1872 Erlangen program. After novel geometries such as hyperbolic and projective geometry had emerged, Klein used group theory to organize them in a more coherent way. Further advancing these ideas, Sophus Lie founded the study of Lie groups in 1884; the third field contributing to group theory was number theory.
Certain abelian group structures had been used implicitly in Carl Friedrich Gauss' number-theoretical work Disquisitiones Arithmeticae, more explicitly by Leopold Kronecker. In 1847, Ernst Kummer made early attempts to prove Fermat's Last Theorem by developing groups describing factorization into prime numbers; the convergence of these various sources into a uniform theory of groups started with Camille Jordan's Traité des substitutions et des équations algébriques. Walther von Dyck introduced the idea of specifying a group by means of generators and relations, was the first to give an axiomatic definition of an "abstract group", in the terminology of the time; as of the 20th century, groups gained wide recognition by the pioneering work of Ferdinand Georg Frobenius and William Burnside, who worked on representation theory of finite groups, Richard Brauer's modular representation theory and Issai Schur's papers. The theory of Lie groups, more locally compact groups was studied by Hermann Weyl, Élie Cartan and many others.
Its algebraic counterpart, the theory of algebraic groups, was first shaped by Claude Chevalley and by the work of Armand Borel and Jacques Tits. The University of Chicago's 1960–61 Group Theory Year brought together group theorists such as Daniel Gorenstein, John G. Thompson and Walter Feit, laying the foundation of a collaboration that, with input from numerous other mathematicians, led to the classification of finite simple groups, with the final step taken by Aschbacher and Smith in 2004; this project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers. Research is ongoing to simplify the proof of this classification; these days, group theory is still a active mathematical branch, impacting many other fields. One of the most familiar groups is the set of integers Z which consists of the numbers... − 4, − 3, − − 1, 0, 1, 2, 3, 4... together with addition. The following properties of integer addition serve as a model for the group axioms given in the definition below.
For any two integers a and b, the sum a + b is an integer. That is, addition of integers always yields an integer; this property is known as closure under addition. For all integers a, b and c, + c = a +. Expressed in words
Joseph Henry Maclagan Wedderburn FRSE FRS was a Scottish mathematician, who taught at Princeton University for most of his career. A significant algebraist, he proved that a finite division algebra is a field, part of the Artin–Wedderburn theorem on simple algebras, he worked on group theory and matrix algebra. His younger brother was the lawyer Ernest Wedderburn. Joseph Wedderburn was the tenth of fourteen children of Alexander Wedderburn of Pearsie, a physician, Anne Ogilvie, he was educated at Forfar Academy in 1895 his parents sent Joseph and his younger brother Ernest to live in Edinburgh with their paternal uncle, J R Maclagan Wedderburn, allowing them to attend George Watson's College. This house was at 3 Glencairn Crescent in the West End of the city. In 1898 Joseph entered the University of Edinburgh. In 1903, he published his first three papers, worked as an assistant in the Physical Laboratory of the University, obtained an MA degree with First Class Honours in mathematics, was elected a Fellow of the Royal Society of Edinburgh, upon the proposal of George Chrystal, James Gordon MacGregor, Cargill Gilston Knott and William Peddie.
Aged only 21 he remains one of the youngest Fellows ever. He studied at the University of Leipzig and the University of Berlin, where he met the algebraists Frobenius and Schur. A Carnegie Scholarship allowed him to spend the 1904–1905 academic year at the University of Chicago where he worked with Oswald Veblen, E. H. Moore, most Leonard Dickson, to become the most important American algebraist of his day. Returning to Scotland in 1905, Wedderburn worked for four years at the University of Edinburgh as an assistant to George Chrystal, who supervised his D. Sc, awarded in 1908 for a thesis titled On Hypercomplex Numbers, he gained a PhD in algebra from the University of Edinburgh in 1908. From 1906 to 1908, Wedderburn edited the Proceedings of the Edinburgh Mathematical Society. In 1909, he returned to the United States to become a Preceptor in Mathematics at Princeton University. Upon the outbreak of the First World War, Wedderburn enlisted in the British Army as a private, he was the first person at Princeton to volunteer for that war, had the longest war service of anyone on the staff.
He served with the Seaforth Highlanders in France, as Lieutenant as Captain of the 10th Battalion. While a Captain in the Fourth Field Survey Battalion of the Royal Engineers in France, he devised sound-ranging equipment to locate enemy artillery, he returned to Princeton after the war, becoming Associate Professor in 1921 and editing the Annals of Mathematics until 1928. While at Princeton, he supervised one of them being Nathan Jacobson. In his years, Wedderburn became an solitary figure and may have suffered from depression, his isolation after his 1945 early retirement was such that his death from a heart attack was not noticed for several days. His Nachlass was destroyed, as per his instructions. Wedderburn received the MacDougall-Brisbane Gold Medal and Prize from the Royal Society of Edinburgh in 1921, was elected to the Royal Society of London in 1933. In all, Wedderburn published about 40 books and papers, making important advances in the theory of rings and matrix theory. In 1905, Wedderburn published a paper that included three claimed proofs of a theorem stating that a noncommutative finite division ring could not exist.
The proofs all made clever use of the interplay between the additive group of a finite division algebra A, the multiplicative group A* = A-. Parshall notes. Meanwhile, Wedderburn's Chicago colleague Dickson found a proof of this result but, believing Wedderburn's first proof to be correct, Dickson acknowledged Wedderburn's priority, but Dickson noted that Wedderburn constructed his second and third proofs only after having seen Dickson's proof. Parshall concludes. A corollary to this theorem yields the complete structure of all finite projective geometry. In their paper on "Non-Desarguesian and non-Pascalian geometries" in the 1907 Transactions of the American Mathematical Society and Veblen showed that in these geometries, Pascal's theorem is a consequence of Desargues' theorem, they constructed finite projective geometries which are neither "Desarguesian" nor "Pascalian". Wedderburn's best-known paper was his sole-authored "On hypercomplex numbers," published in the 1907 Proceedings of the London Mathematical Society, for which he was awarded the D.
Sc. the following year. This paper gives a complete classification of semisimple algebras, he showed that every semisimple algebra finite-dimensional can be constructed as a direct sum of simple algebras and that every simple algebra is isomorphic to a matrix algebra for some division ring. The Artin–Wedderburn theorem generalises this result, with the ascending chain condition, his best known book is his Lectures on Matrices, which Jacobson praised as follows: That this was the result of a number of years of painstaking labour is evidenced by the bibliography of 661 items covering the period 1853 to 1936. The work is, not a compilation of the literature, but a synthesis, Wedderburn's own, it contains a number of original contributions to the subject. About Wedderburn's teaching: He was a shy man and much preferred looking at the blackboard to looking at the students, he had the galley proof
Not to be confused with Paul Cohen. Paul Moritz Cohn FRS was Astor Professor of Mathematics at University College London, 1986-9, author of many textbooks on algebra, his work was in the area of algebra non-commutative rings. He was the only child of Jewish parents, James Cohn, owner of an import business, Julia, a schoolteacher. Both of his parents were born in Hamburg, his ancestors came from various parts of Germany. His father fought in the German army in World War I. A street in Hamburg is named in memory of his mother; when he was born, his parents were living with his mother's mother in Isestraße. After her death in October 1925, the family moved to a rented flat in a new building in Lattenkamp, in the Winterhude quarter, he attended a kindergarten in April 1930, moved to Alsterdorfer Straße School. After a while, he had a new teacher, a National Socialist, who picked on him and punished him without cause, thus in 1931, he moved to the Meerweinstraße School. Following the rise of the Nazis in 1933, his father's business was confiscated and his mother dismissed.
He moved to a Jewish school. In mid-1937, the family moved to Klosterallee; this was nearer the synagogue and other pupils, being in the Jewish area. His German teacher was the son of the poet Jakob Loewenberg. On the night of 9/10 November 1938, his father was arrested and sent to Sachsenhausen concentration camp, he was told to emigrate. Cohn went to Britain in May 1939 on the Kindertransport to work on a chicken farm, never saw his parents again, he corresponded with them until late 1941. At the end of the War, he learned that they were deported to Riga on 6 December 1941 and never returned. At the end of 1941, the farm closed, he acquired a work permit and worked in a factory for 4 1/2 years. He passed the Cambridge Scholarship Examination, won an exhibition to Trinity College, Cambridge, he received a B. A in Mathematics from Cambridge University in 1948 and a Ph. D. in 1951. He spent a year as a Chargé de Recherches at the University of Nancy. On his return, he became a lecturer in mathematics at Manchester University.
He was a visiting professor at Yale University in 1961–62, for part of 1962 was at the University of California at Berkeley. On his return, he became Reader at Queen Mary College, he was a visiting professor at the University of Chicago in 1964 and at the State University of New York at Stony Brook in 1967. By he was regarded as one of the world's leading algebraists. In 1967, he became head of the Department of Mathematics at Bedford College, he held several visiting professorships, in America, Delhi, Canada and Bielefeld. He was awarded the Lester R Ford Award from the Mathematical Association of America in 1972 and the Senior Berwick Prize of the London Mathematical Society in 1974. In the early 1980s, funding cuts caused the closure of the small colleges of the University of London. Cohn moved to University College in 1984, together with the two other experts at Bedford on ring theory, Bill Stephenson and Warren Dicks, he became Astor Professor of Mathematics there in 1986. He continued to be a visiting professor, for example to the University of Alberta in 1986 and to Bar Ilan University in 1987.
He retired in 1989, but remained active as Professor Emeritus and Honorary Research Fellow until his death. He was President of the London Mathematical Society, 1982-4, having been its secretary, 1965–67 and a Council member in 1968–71, 1972–75 and 1979–82, he was editor of the Society's Monographs in 1968–77 and 1980–93. He was elected a Fellow of the Royal Society in 1980 and was on its council, 1985–87, he was a member of the Mathematical Committee of the Science Research Council, 1977–1980. He chaired the National Committee for Mathematics, 1988-9. In all, Cohn wrote nearly 200 mathematical papers, he worked in many areas of algebra in non-commutative ring theory. His first papers, covering many topics, were published in 1952, he generalised a theorem due to Wilhelm Magnus, worked on the structure of tensor spaces. In 1953 he published a joint paper with Kurt Mahler on pseudo-valuations and in 1954 he published a work on Lie algebras. Papers over the next few years covered areas such as group theory, field theory, Lie rings, Abelian groups and ring theory.
He published his first book, Lie groups, in 1957. After that, he moved into the areas of Jordan algebras, Lie division rings, skew fields, free ideal rings and non-commutative unique factorisation domains, he published his second book, Linear equations, in 1958 and his third, Solid geometry, in 1961. Universal algebra appeared in 1965. After that, he concentrated on the theory of algebras, his monograph Free rings and their relations appeared in 1971. It covered the work of Cohn and others on free associative algebras and related classes of rings free ideal rings, he included all of his own published results on the embedding of rings into skew fields. The second, enlarged edition appeared in 1985. Cohn wrote undergraduate textbooks. Algebra volume I appeared in 1974 and volume II in 1977; the second edition, in three volumes, was published by Wiley between 1982 and 1991. These volumes were in line with the British curricula at the time and include both linear algebra and abstract algebra. Cohn wrote a subsequent revised iteration the first volume as Classical Algebra as a more "user friendly" version for undergraduates (
Tsit Yuen Lam
Tsit Yuen Lam is a Hong Kong-American mathematician specializing in algebra ring theory and quadratic forms. Lam earned his bachelor's degree at the University of Hong Kong in 1963 and his Ph. D. at Columbia University in 1967 under Hyman Bass, with a thesis titled On Grothendieck Groups. Subsequently he was an instructor at the University of Chicago and since 1968 he has been at the University of California, where he became assistant professor in 1969, associate professor in 1972, full professor in 1976, he served as assistant department head several times. From 1995 to 1997 he was Deputy Director of the Mathematical Sciences Research Institute in Berkeley, California. Among his doctoral students is Richard Elman. From 1972 to 1974 he was a Sloan Fellow. In 1982 he was awarded the Leroy P. Steele Prize for his textbooks. In 2012 he became a fellow of the American Mathematical Society. Serre’s Conjecture. Lecture Notes in Mathematics, Springer, 1978 Serre’s Problem on Projective Modules. Springer 2006.
Benjamin 1973, 1980. Graduate Texts in Mathematics, Springer 1991, 2nd edition 2001, ISBN 0-387-95325-6 Lectures on Modules and Rings. Springer, Graduate Texts in Mathematics 1999, ISBN 978-0-387-98428-5 Orderings and Quadratic Forms. AMS 1983 Exercises in Classical Ring Theory. Springer 1985 Representations of Finite Groups: A Hundred Years. Part I, Part II. Notices of the AMS 1998. Lam's homepage