Celebratio Mathematica

J. von Neumann and M. H. Stone: “The determination of representative elements in the residual classes of a Boolean algebra,” Fundam. Math. 25 (1935), pp. 353–378. JFM 61.0976.02 Zbl 0012.24403 article

If \( A \) is any abstract ring and \( \mathfrak{a} \) is a left- and right-ideal in \( A \), we may consider the problem of selecting a single representative element from each of the residual classes \( (\operatorname{mod} \mathfrak{a}) \) in such a manner that sums and products of representative elements are themselves representative elements. If such a selection is possible, the representative elements evidently constitute a subsystem of \( A \) which is isomorphic to the quotient-ring \( A/\mathfrak{a} \) under the correspondence carrying each residual class \( (\operatorname{mod} \mathfrak{a}) \) into its representative element. In this paper we shall confine our attention to rings in which every element is idempotent — that is, in which the law \( aa = a \) obtains. These rings will be seen to have the formal properties of certain algebras of classes, and will therefore be termed Boolean rings. A particular case of the representation problem for Boolean rings has previously been discussed by one of us [von Neumann 1931]. Here we shall examine the problem on an abstract basis, giving sufficient conditions for the existence of a solution, special cases in which no solution exists, and special cases in which a solution exists if and only if \[ \aleph_{n+1} = 2^{\aleph_n} .\] The sufficient conditions given here can be applied to the particular case mentioned above.

G. W. Mackey: “Marshall H. Stone: Mathematician, statesman, advisor, and friend,” pp. 15–25 in Operator algebras, quantization, and noncommutative geometry: A centennial celebration honoring John von Neumann and Marshall H. Stone (Baltimore, MD, 15–16 January 2003). Edited by R. S. Doran and R. V. Kadison. Contemporary Mathematics 365. American Mathematical Society (Providence, RI), 2004. MR 2106814 Zbl 1059.01539 incollection

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Operator algebras, quantization, and noncommutative geometry: A centennial celebration honoring John von Neumann and Marshall H. Stone (Baltimore, MD, 15–16 January 2003). Edited by R. S. Doran and R. V. Kadison. Contemporary Mathematics 365. American Mathematical Society (Providence, RI), 2004. MR 2106812 Zbl 1056.58001 book

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