Boolean ring

A Boolean ring is a ring with idempotent multiplication, that is, a ring $(R,+,\cdot )$ ${\ displaystyle (R, +, \ cdot)}$ $(R, +, \ cdot)$ , wherein $x^{2}=x$ ${\ displaystyle x ^ {2} = x}$ $x ^ {2} = x$ for all $x\in R$ ${\ displaystyle x \ in R}$ $x \ in R$ ^[1] ^[2] ^[3] .

Content

Connection with Boolean Algebra

The most famous example of a Boolean ring is obtained from Boolean algebra $(B,\land ,\lor ,\lnot )$ ${\ displaystyle (B, \ land, \ lor, \ lnot)}$ the introduction of addition and multiplication as follows:

$x+y=(x\land \lnot y)\lor (\lnot x\land y)$ ${\ displaystyle x + y = (x \ land \ lnot y) \ lor (\ lnot x \ land y)}$ ,
$x\cdot y=x\land y$ ${\ displaystyle x \ cdot y = x \ land y}$ .

In particular, a Boolean of some set $X$ ${\ displaystyle X}$ forms a Boolean ring with respect to the symmetric difference and intersection of the subsets . In this regard, the main example introducing addition in the Boolean ring as “ exclusive or ” for Boolean algebras, and multiplication as a conjunction , sometimes the symbol is used for addition in Boolean rings $\oplus$ ${\ displaystyle \ oplus}$ , and for multiplication - signs of the lattice lower bound ( $\land$ ${\ displaystyle \ land}$ , $\cap$ ${\ displaystyle \ cap}$ , $\sqcap$ ${\ displaystyle \ sqcap}$ )

Every Boolean ring obtained in this way from Boolean algebra has a unit coinciding with the unit of the original Boolean algebra. In addition, every Boolean ring with unit $(R,+,\cdot ,1)$ ${\ displaystyle (R, +, \ cdot, 1)}$ uniquely defines a Boolean algebra with the following definitions of operations:

$x\land y=x\cdot y$ ${\ displaystyle x \ land y = x \ cdot y}$ ,
$x\lor y=x+y+xy$ ${\ displaystyle x \ lor y = x + y + xy}$ ,
$\lnot x=1+x$ ${\ displaystyle \ lnot x = 1 + x}$ .

Properties

In every boolean ring $(R,+,\cdot )$ ${\ displaystyle (R, +, \ cdot)}$ done $x+x=0$ ${\ displaystyle x + x = 0}$ as a consequence of idempotency with respect to multiplication:

x+x=(x+x)^{2}=x+x+x+x

{\ displaystyle x + x = (x + x) ^ {2} = x + x + x + x}

,

and since $(R,+)$ ${\ displaystyle (R, +)}$ in the ring is an abelian group , then we can subtract the component $x+x$ ${\ displaystyle x + x}$ from both sides of this equation.

Every Boolean ring is commutative , which is also a consequence of the idempotency of multiplication:

x+y=(x+y)^{2}=x^{2}+xy+yx+y^{2}=x+y+xy+yx

{\ displaystyle x + y = (x + y) ^ {2} = x ^ {2} + xy + yx + y ^ {2} = x + y + xy + yx}

,

what gives $xy+yx=0$ ${\ displaystyle xy + yx = 0}$ , which in turn means $xy=yx$ ${\ displaystyle xy = yx}$ .

Every nontrivial finite Boolean ring is a direct sum of residue fields modulo 2 ( $\mathbb {Z} _{2}$ ${\ displaystyle \ mathbb {Z} _ {2}}$ ) and has a unit .

Factor ring $R/I$ ${\ displaystyle R / I}$ any Boolean ring by an arbitrary ideal $I$ ${\ displaystyle I}$ also a boolean ring. In the same way, any subring of some Boolean ring is a Boolean ring. Every simple ideal $P$ ${\ displaystyle P}$ in a boolean ring $R$ ${\ displaystyle R}$ is maximum : factor ring $R/P$ ${\ displaystyle R / P}$ is a domain of integrity , as well as a Boolean ring, therefore it is isomorphic to the field $\mathbb {F} _{2}$ ${\ displaystyle \ mathbb {F} _ {2}}$ that shows the maximum $P$ ${\ displaystyle P}$ . Since maximal ideals are always simple, the concepts of simple and maximal ideals coincide for Boolean rings.

Boolean rings are absolutely flat , that is, any module above them is flat .

Every finite ideal of a Boolean ring is central .

Notes

↑ Freilly, 1976 , p. 200.
↑ Gerstein, 1964 , p. 91.
↑ McCoy, 1968 , p. 46.

Literature

M. Atiyah , IG Macdonald . Introduction to Commutative Algebra. - Westview Press, 1969. - ISBN 978-0-201-40751-8 .
John B. Fraleigh. A First Course In Abstract Algebra. - 2nd. - Reading: Addison-Wesley , 1976. - ISBN 0-201-01984-1 .
IN Herstein. Topics in Algebra. - Waltham: Blaisdell Publishing, 1964 .-- ISBN 978-1114541016 .
Neal H. McCoy. Introduction To Modern Algebra. - revised. - Boston: Allyn and Bacon, 1968.

[_7738f0da6e7c2dca-1] Freilly, 1976 , p. 200.

[_b582e01b4e35068d-2] Gerstein, 1964 , p. 91.

[_9d6cb46b9a53f562-3] McCoy, 1968 , p. 46.