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An associative ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016980/b0169801.png" /> whose elements are all idempotent, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016980/b0169802.png" /> for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016980/b0169803.png" />. Any Boolean ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016980/b0169804.png" /> is commutative and is a subdirect sum of fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016980/b0169805.png" /> of two elements, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016980/b0169806.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016980/b0169807.png" />. A finite Boolean ring <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016980/b0169808.png" /> is a direct sum of fields <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016980/b0169809.png" /> and therefore has a unit element.
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An associative ring $K$ whose elements are all idempotent, i.e. $x^2=x$ for any $x\in K$. Any Boolean ring $K\neq0$ is commutative and is a subdirect sum of fields $\mathbf Z_2$ of two elements, and $x+x=0$ for all $x\in K$. A finite Boolean ring $K\neq0$ is a direct sum of fields $\mathbf Z_2$ and therefore has a unit element.
  
 
A Boolean ring is the ring version of a [[Boolean algebra|Boolean algebra]], namely: Any Boolean algebra is a Boolean ring with a unit element under the operations of addition and multiplication defined by the rules
 
A Boolean ring is the ring version of a [[Boolean algebra|Boolean algebra]], namely: Any Boolean algebra is a Boolean ring with a unit element under the operations of addition and multiplication defined by the rules
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016980/b01698010.png" /></td> </tr></table>
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$$(x+y)=(x\cap Cy)\cup(y\cap Cx),\quad x\cdot y=x\cap y,$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016980/b01698011.png" /> is the complement of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016980/b01698012.png" />. The zero and the unit of the ring are the same as, respectively, the zero and the unit of the algebra. Conversely, every Boolean ring with a unit element is a Boolean algebra under the operations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016980/b01698013.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016980/b01698014.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016980/b01698015.png" />.
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where $Cx$ is the complement of $x$. The zero and the unit of the ring are the same as, respectively, the zero and the unit of the algebra. Conversely, every Boolean ring with a unit element is a Boolean algebra under the operations $x\cup y=x+y+xy$, $x\cap y=x\cdot y$, $Cx=1+x$.
  
 
====References====
 
====References====
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====Comments====
 
====Comments====
The operation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b016/b016980/b01698016.png" /> is known as the symmetric difference. Think of the Boolean algebra of all subsets of a given set under union, intersection and complement to interpret these formulas.
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The operation $x+y=(x\cap Cy)\cup(y\cap Cx)$ is known as the symmetric difference. Think of the Boolean algebra of all subsets of a given set under union, intersection and complement to interpret these formulas.
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Rudeanu,  "Boolean functions and equations" , North-Holland  (1974)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S. Rudeanu,  "Boolean functions and equations" , North-Holland  (1974)</TD></TR></table>

Latest revision as of 15:09, 13 August 2014

An associative ring $K$ whose elements are all idempotent, i.e. $x^2=x$ for any $x\in K$. Any Boolean ring $K\neq0$ is commutative and is a subdirect sum of fields $\mathbf Z_2$ of two elements, and $x+x=0$ for all $x\in K$. A finite Boolean ring $K\neq0$ is a direct sum of fields $\mathbf Z_2$ and therefore has a unit element.

A Boolean ring is the ring version of a Boolean algebra, namely: Any Boolean algebra is a Boolean ring with a unit element under the operations of addition and multiplication defined by the rules

$$(x+y)=(x\cap Cy)\cup(y\cap Cx),\quad x\cdot y=x\cap y,$$

where $Cx$ is the complement of $x$. The zero and the unit of the ring are the same as, respectively, the zero and the unit of the algebra. Conversely, every Boolean ring with a unit element is a Boolean algebra under the operations $x\cup y=x+y+xy$, $x\cap y=x\cdot y$, $Cx=1+x$.

References

[1] M.H. Stone, "The theory of representations for Boolean algebras" Trans. Amer. Math. Soc. , 40 (1936) pp. 37–111
[2] I.I. Zhegalkin, "On the technique of computation of propositions in symbolic logic" Mat. Sb. , 34 : 1 (1927) pp. 9–28 (In Russian) (French abstract)
[3] D.A. Vladimirov, "Boolesche Algebren" , Akademie Verlag (1978) (Translated from Russian)
[4] R. Sikorski, "Boolean algebras" , Springer (1969)


Comments

The operation $x+y=(x\cap Cy)\cup(y\cap Cx)$ is known as the symmetric difference. Think of the Boolean algebra of all subsets of a given set under union, intersection and complement to interpret these formulas.

References

[a1] S. Rudeanu, "Boolean functions and equations" , North-Holland (1974)
How to Cite This Entry:
Boolean ring. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boolean_ring&oldid=32903
This article was adapted from an original article by Yu.M. Ryabukhin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article