Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [boolean-ring]

Use this tag for questions related to Boolean rings such as the ring of integers modulo $2$ $\mathbb Z/2\mathbb Z$.

A Boolean ring R is a ring for which $x^2 = x$ for all $x\in R$. That is, $R$ consists of idempotent elements only.

Every Boolean ring gives rise to a Boolean algebra with ring multiplication corresponding to conjunction or meet ∧, and ring addition to exclusive disjunction or symmetric difference (not disjunction ∨, which would constitute a semiring).

Examples of Boolean rings include

  • $\mathbb Z/2\mathbb Z$ (AND is multiplication, XOR is addition)
  • the power set of any set X where addition is symmetric difference, and multiplication is intersection, and
  • the set of all finite or co-finite subsets of X, again with symmetric difference and intersection as operations.

More generally, with the operations of the last two examples, any field of sets is a Boolean ring. By Stone's representation theorem, every Boolean ring is isomorphic to a field of sets treated as a ring with those operations.

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Power Set of $X$ is a Ring with Symmetric Difference, and Intersection

I'm studying for an abstract algebra exam and one of the review questions was this: Let $X$ be a set, and $\mathcal P(X)$ be the power set of $X$. Consider the operations $\Delta$ = symmetric difference (a.k.a. "XOR"), and $\bigcap$ =…
Max
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A Finite Boolean Ring is Generated by Finitely Many Copies of $\mathbb{Z} / 2 \mathbb{Z}$

I'm trying to prove that every finite Boolean ring, $R$, is isomorphic to a finite number of copies of $\mathbb{Z} / 2 \mathbb{Z}$: \begin{align} R \cong \mathbb{Z} / 2 \mathbb{Z} \; \times \cdots \; \times \; \mathbb{Z} / 2…
Junaid Aftab
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An $m$-ary function that represents all $n$-ary functions

Motivation It is well-known that any binary operator $*$ on the boolean ring $\{0,1\}$ can be represented using only one of the $\operatorname{NAND}$ and $\operatorname{NOR}$ operators. For example, $$x\rightarrow…
user593746
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Ring of Sets vs Ring in Universal Algebra

I actually want to continue this post. I believe the naming convention in Mathematics is consistent, such that there are no clearly distinguish objects have the same name. However, I'm not sure how to relate the Ring of Sets and the Ring in…
Abe Vallerian
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Help understanding why finite Boolean rngs must be rings

I'm working through the exercises in Introduction to Boolean Algebra by Halmos and Givant. Looking to show the following, an exercise from the first chapter: every finite Boolean rng must have a unit. Halmos and Givant attribute this observation to…
omegaplusone
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Prime ideal $\implies$ maximal in a Boolean ring

I want to show that a prime ideal in a non-unital Boolean ring $B$ is maximal ideal. If the ring contains unity then it is easy. As Boolean rings are commutative, for a prime ideal $P$ the ring $B/P$ is both integral domain aswell as Boolean ring.…
Infinity_hunter
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A finite subset of a Boolean algebra, generates a finite subalgebra.

Let $\mathscr{B}$ be a Boolean algebra and let $F$ be a finite subset of $\mathscr{B}$. Is $\left$, the subalgebra generated by $F$, necessarily a finite subalgebra of $\mathscr{B}$? I thinks that this is true based on following argument:…
Dilemian
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Jacobson Radical of a Boolean Ring is Zero

I am trying to prove that the Jacobson radical $J(R)$ of a Boolean ring $R$ is $\{0\}$, even when $R$ does not have a unity. In the case where the ring has an identity element, the proof is straightforward. However, I encounter difficulties…
Ayrton Porto
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Finite Commutative ring with 100 elements where $x^2=x$?

Does there exist a finite Commutative ring with 100 elements where $x^2=x$ for every $x\in R$? I know finite Boolean rings has the property this property but they have cardinality $2^n$, for some $n$. Also Boolean rings are rings with identity. Does…
Learner
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A property of Boolean algebra

In a Boolean algebra $\mathcal B$, we know that $$x+\bar{x}y=x+y\text{ for all } x, y\in \mathcal B.$$ By following the above identity, we can also write $$xy+\bar{x}yz=xy+yz.$$ Can we write $$\bar{y}xz+yp=xz+yp,\text{ where $p$ is distinct from $x$…
gete
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Rings in which each element is a sum of $n$ commuting idempotents

Let $n$ be a nonnegative integer. Let $R$ be a nonunital ring such that every element of $R$ is a sum of $n$ pairwise commuting idempotents. (As usual, the class of nonunital rings includes the class of unital rings.) In Theorem 1 of…
darij grinberg
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Understanding boolean rings

I'm reading a book "Introduction to Abstract Algebra" by Neal McCoy. I've come across a few exercises which discuss "Boolean rings". The text defines Boolean rings as: A ring $R$ is a Boolean ring if $a^2 = a$ for every element $a$ of $R$. and…
bt26
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Calculating Walsh Hadamard Transform

Can anyone say me the steps involved in calculating Walsh Hadamard Transform $(W_f(a))$ for a Boolean function $f(x)=x^3$ in a finite field $GF=F_4$?
user378779
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Boolean Ring has $2^N$ elements "proof"

I was thinking about a proof that all finite booleans rings has $2^N$ elements. My attempt was the following: Consider a boolean ring and all of its isolated elements, i.e., elements that isn't a sum or product of other elements and are differents…
Carinha logo ali
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A boolean ring which is local must be isomorphic to $\Bbb{F}_2$.

Recall: a Boolean ring is a (commutative) ring $R$ where $\forall x \in R: x^2=x$. I don't really know how to proceed. I have tried some things like If $x,y \ne 0$ in $R$ such that $x^2=x$ and $y^2=y$, then by the fact that $R$ is local, we must…
soggycornflakes
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