# Boolean ring

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## Boolean ring

[¦bül·ē·ən ′riŋ]
(mathematics)
A commutative ring with the property that for every element a of the ring, a × a and a + a = 0; it can be shown to be equivalent to a Boolean algebra.
McGraw-Hill Dictionary of Scientific & Technical Terms, 6E, Copyright © 2003 by The McGraw-Hill Companies, Inc.
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Theorem.2.3 Let R be a finite Boolean ring.Then [absolute value of ([R.sup.x])] = 1.
It follows that the proper subset A, a maximal set of B forms a Boolean ring. B is a Boolean-near-ring, whose proper subset is a Boolean-ring, then by definition, B is a SmarandacheBoolean-near-ring.
Hence A is a maximal set with uni-element and by theorem 1 and definition A, a maximal set of B forms a Boolean ring.
Ryabukhin, "Boolean ring," in Encyclopaedia of Mathematics, M.
In the literature such rings exist naturally, for instance, the rings [Z.sub.6] (modulo integers), [Z.sub.10] (modulo integers), Boolean ring.

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