Nathan Jacobson

The Theory of Rings

The book is mainly concerned with the theory of rings in which both maximal and minimal conditions hold for ideals (except in the last chapter, where rings of the type of a maximal order in an algebra are considered). The central idea consists of representing rings as rings of endomorphisms of an additive group, which can be achieved by means of the regular representation.

Jordan homomorphisms of rings

The primary aim of this paper is to study mappings J of rings that are additive and that satisfy the conditions

Some Remarks on One-Sided Inverses

{\left( {{a^2}} \right)^J} = {\left( {{a^J}} \right)^2},\;{\left( {aba} \right)^J} = {a^J}{b^J}{a^J}

Completely reducible Lie algebras of linear transformations

(1) Such mappings will be called Jordan homomorphisms. If the additive groups admit the operator 1/2 in the sense that 2x = a has a unique solution (1/2)a for every a, then conditions (1) are equivalent to the simpler condition

Classification and representation of semi-simple Jordan algebras

{\left( {ab} \right)^J} + {\left( {ba} \right)^J} = {a^J}{b^J} + {b^J}{a^J}

A Note on Lie Algebras of Characteristic p

(2) Mappings satisfying (2) were first considered by Ancochea [1], [2](1). The modification to (1) is essentially due to Kaplansky [13]. Its purpose is to obviate the necessity of imposing any restriction on the additive groups of the rings under consideration.