Ring (mathematics)

About: Ring (mathematics) is a research topic. Over the lifetime, 19980 publications have been published within this topic receiving 233849 citations. The topic is also known as: ring possibly without identity.

A Taste of Jordan Algebras

Abstract: In this book, Kevin McCrimmon describes the history of Jordan Algebras and he describes in full mathematical detail the recent structure theory for Jordan algebras of arbitrary dimension due to Efim Zel'manov. To keep the exposition elementary, the structure theory is developed for linear Jordan algebras, though the modern quadratic methods are used throughout. Both the quadratic methods and the Zelmanov results go beyond the previous textbooks on Jordan theory, written in the 1960's and 1980's before the theory reached its final form. This book is intended for graduate students and for individuals wishing to learn more about Jordan algebras. No previous knowledge is required beyond the standard first-year graduate algebra course. General students of algebra can profit from exposure to nonassociative algebras, and students or professional mathematicians working in areas such as Lie algebras, differential geometry, functional analysis, or exceptional groups and geometry can also profit from acquaintance with the material. Jordan algebras crop up in many surprising settings and can be applied to a variety of mathematical areas. Kevin McCrimmon introduced the concept of a quadratic Jordan algebra and developed a structure theory of Jordan algebras over an arbitrary ring of scalars. He is a Professor of Mathematics at the University of Virginia and the author of more than 100 research papers.

On rings of operators

Abstract: Let 5C be a complex Hubert Space, J3(3C) the ring of bounded operators on 3C, E an abelian symmetric subring of B(3Z) containing the identity which is closed in the weak operator topology, E\ the commutant of E, and suppose E\ has a cyclic vector £o which we normalize so that |£o| = 1 . Diximier [ l] has shown that E (respect. Ei), as a Banach space, is the dual of the Banach space R (respect. Ri) of all linear forms on E (respect. E\) that are continuous in the ultra-strong topology of E (respect. Ei). In this note we show that every TCzR is also continuous in the weak operator topology of E, from which it follows that a linear functional T on E is continuous in either the weak, ultraweak, strong, or ultrastrong topologies if and only if it is continuous in all four simultaneously. In the process, we obtain an integral representation for such T, which we later use in a theorem on centrally reducible positive functionals on E%. We denote the maximal ideal space of E by M, and for A, B, • • • £ E , we denote the corresponding Gel'fand transforms by a, 6, • • • . Then A—>a is an isometric isomorphism from £ onto C(M). Consequently, every bounded linear functional on E has the form

Cohen-Macaulay Rings, Invariant Theory, and the Generic Perfection of Determinantal Loci

Abstract: 0. Introduction. Let R be a commutative Noetherian ring with identity and let I be a proper ideal of R. A classical result of Krull is that if I can be generated by r elements then the rank or altitude of I (the greatest rank of any minimal prime of I) is at most r. If, moreover, the grade of I (the length of the longest R-sequence contained in I) is r, then I enjoys certain special properties summarized in the term "perfect" as used by iRees [30, p. 32]: I is perfect if the homological (or projective) dimension of R/I as an R-module is equal to the grade of I. The associated primes of a perfect ideal I all have the same grade as I, that is, perfect ideals are grade unmixed. If R is Cohen-Macaulay, the grade of any ideal is equal to its little rank of height (the least rank of any minimal prime) ; in particular, the notions of grade and rank coincide on -primes, and perfect ideals are rank unmixed. Moreover, if I is perfect in a Cohen-Macaulay ring R, R/I is again (Cohen-Macaulay. Macaulay's famous theorem that in a polynomial ring over a field a rank r ideal which can be generated by r elements is rank unmixed [36, p. 203] is then a consequence of two facts: a polynomial ring over a field is CohenMacaulay, and a grade r ideal generated by r elements is perfect. This is the classical example of a perfect ideal. Good discussions of the subject. are available: see [9], [24, ? 25], [30], [18, Ch. 3], and [36, Appendix 6]. The Noetherian restriction on R is, for certain purposes, unnecessary in the discussion of perfect ideals, if one adopts a suitable definition of grade. This idea is worked out in [1]. Suppose that R is (locally) regular, and I is an ideal of R such that R/I is not the direct product of two rings in a nontrivial way. Then I is perfect if and only if R/I is Cohen-Macaulay. In particular, this is the situation when R is a polynomial ring over a field and I is homogeneous. It is very natural, then, to hunt for perfect ideals. Relatively few classes are known, but several authors [4, 6, 8, 33] have established the perfection