Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2024 MCQ for Transactions of the American Mathematical Society is 1.48 .

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.

 

Noncommutative Jordan rings
HTML articles powered by AMS MathViewer

by Kevin McCrimmon
Trans. Amer. Math. Soc. 158 (1971), 1-33
DOI: https://doi.org/10.1090/S0002-9947-1971-0310024-7

Abstract:

Heretofore most investigations of noncommutative Jordan algebras have been restricted to algebras over fields of characteristic $\ne 2$ in order to make use of the passage from a noncommutative Jordan algebra $\mathfrak {A}$ to the commutative Jordan algebra ${\mathfrak {A}^ + }$ with multiplication $x \cdot y = \frac {1}{2}(xy + yx)$. We have recently shown that from an arbitrary noncommutative Jordan algebra $\mathfrak {A}$ one can construct a quadratic Jordan algebra ${\mathfrak {A}^ + }$ with a multiplication ${U_x}y = x(xy + yx) - {x^2}y = (xy + yx)x - y{x^2}$, and that these quadratic Jordan algebras have a theory analogous to that of commutative Jordan algebras. In this paper we make use of this passage from $\mathfrak {A}$ to ${\mathfrak {A}^ + }$ to derive a general structure theory for noncommutative Jordan rings. We define a Jacobson radical and show it coincides with the nil radical for rings with descending chain condition on inner ideals; semisimple rings with d.c.c. are shown to be direct sums of simple rings, and the simple rings to be essentially the familiar ones. In addition we obtain results, which seem to be new even in characteristic $\ne 2$, concerning algebras without finiteness conditions. We show that an arbitrary simple noncommutative Jordan ring containing two nonzero idempotents whose sum is not 1 is either commutative or quasiassociative.
References
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 17A15
  • Retrieve articles in all journals with MSC: 17A15
Bibliographic Information
  • © Copyright 1971 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 158 (1971), 1-33
  • MSC: Primary 17A15
  • DOI: https://doi.org/10.1090/S0002-9947-1971-0310024-7
  • MathSciNet review: 0310024