Questions tagged [boolean-ring]

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

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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 $B/P$ ...
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The Boolean ring $\mathscr{P}(\Bbb N)$ has an ideal $\mathscr{P}(\Bbb N_+)$. Find the elements of $\mathscr{P}(\Bbb N)/\mathscr{P}(\Bbb N_+)$.

Denote $\Bbb N_+ = \Bbb N \setminus\{0\}$. The Boolean ring $\mathscr{P}(\Bbb N)$ has an ideal $\mathscr{P}(\Bbb N_+)$. Find the elements of $\mathscr{P}(\Bbb N)/\mathscr{P}(\Bbb N_+)$ and compute the ...
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Are Boolean rings integral domains? [duplicate]

Are Boolean rings integral domains? My assumption was no. The "product" in Boolean rings is the intersection $\cap$ of two sets from $\mathscr{P}(X)$. If Boolean rings we're integral ...
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Ring with every element idempotent

I am having query in this MCQ. Let $R$ be a ring such that every element is idempotent then (a) Every prime ideal is maximal ideal. (b) Every maximal ideal is prime ideal (c) if $|R|> 2$ implies $...
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2 answers
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Proving from De Morgan's laws

I was proving: $J.K'.L+J.K.L'+JKL=J.(L+K)$ I have done the following: From LHS: $J.K'.L+J.K.L'+JKL$ $=J.(K'.L+K.L'+K.L)$ $=J.(K'.L+K.(L'+L))$ Since $L'+L=1:$ $=J.(K'L+K)$ [' is complement] Now I'm ...
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Boolean ring prime ideals [duplicate]

I'm asked 2 question: 1- Prove or disprove: In every Boolean ring prime ideals are maximal. 2- Prove or disprove: If $R$ is Boolean and $p$ is a prime ideal of $R$, then the ring homomorphism $R\to ...
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A question concerning supremum in a Boolean ring

In a Boolean $\sigma$-ring, is it true that for every sequence {$x_n$} and y, we have $supx_n\cdot y = sup(x_n\cdot y)$ ? Background: I was reading 'Measure Theory' written by Paul R.Halmos (in the ...
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Number of truth outputs of a Boolean function

Disjunctive Normal Form (DNF) A Boolean function $f$ of $n$ variables is said to be in a DNF if it is a disjunction of conjunction in $n$ variables. That is, each conjunction includes all the ...
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Number of terms in a DNF equals the number of truth outputs

In a Boolean function $f$, it seems that the number of terms in a Disjunctive Normal Form (DNF) is equal to the number of truth outputs of $f$. For example, If $f=A+B$, then clearly, $f$ has $2^2-1=3$ ...
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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}...
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Dixon/quadratic sieve factoring with Gaussian elimination

Given an integer $N$ to be factorized, seek $k+1$ integers $a_i$ such that the rest of $a_i^2$ divided with $N$ is a $k$-smooth integer $s_i$. The task is to multiply $s_{i_1}\cdots s_{i_m}$ such ...
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commutative ring to boolean algebra

Let X be a set. We know that $(P(X),\triangle, \cap )$ is a commutative ring with the zero-element $ \emptyset $ and the one-element $X$. $P(X)$ is the power set, $\triangle$ the symmetrical ...
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Define an isomorphism between finite boolean ring and $\mathbb{Z}_2\times \cdots\times \mathbb{Z}_2$

I'm trying to prove any finite boolean ring $R$ is isomorphic to $\mathbb{Z}_2\times\cdots\times\mathbb{Z}_2$. I know this question is already on this site, but the proofs I saw used the ring's ...
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Is Zero ring a Boolean ring?

I have a simple doubt. Is zero ring a Boolean ring? ( We do have $0 ^2 =0$ ). Or do we assume a Boolean ring to be non-zero?. The doubt came in my mind while showing that in a Boolean ring every prime ...
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What is the algebraic structure of all the quadtrees under these operations?

I was implementing a quadtree in a programming language, and I realized I could define operations such as negation $\bar{x}$, reunion $a\cup b$ and intersection $a \cap b$ on these objects. These ...
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Identify $S = 0$ for any set $S$ and form a non-boolean ring out of $\Delta$ on sets?

Let $S$ be a set and define for each $A \subset S$ the negative of $A$ to be $-A = S \setminus A$. So that $A + S\setminus A = S$. Thus we identify $0 = S$. The $+$ operation is still $\Delta$ so ...
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Show that atom of Boolean algebra B inside B' is also the atom of B' where B' is subalgebra of B. [closed]

Let B := (B, ≤, ∨, ∧, c , 0, 1) be a Boolean algebra and B' := (B' , ≤, ∨, ∧, c , 0, 1) be a Boolean subalgebra of B. Show that an element of B0 that is an atom of B must also be an atom of B' . ...
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What is the motivation behind defining $[\alpha] + [\beta] = [(\alpha \land \lnot \beta) \lor (\lnot \alpha \land \beta)]$?

Consider $\mathcal{F}$ to be the set of all well-formed formulae (wffs), and $\mathcal{F}/\equiv$ denotes the set of equivalence classes under the equivalence relation induced by $\equiv$ (logical ...
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Proof verification: prime ideals in Boolean rings

This question have been answered many times but I didn't find a similar proof, I submit it here for verification. Let $R$ be a Boolean ring (i.e. $\forall r\in R, \exists n>1: r^n=r$). Prove that ...
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Boolean rings are associative $\mathbb{Z}/2\mathbb{Z}$-algebras

Let $B$ be a boolean ring, that is, for all $x\in B$ we have $x^2 = x$. Then we know a few things about them. First $B$ has characteristic $2$, since taking $x\in B$ we have $$x+x = (x+x)^2 = x^2 + x^...
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Given a boolean matrix $M$ what are the matrices formed by replacing $1s$ in $M$ with $0s$ called?

Given two boolean matrices $A$ and $B$ over some common dimensions one can form an order via $A\leq B\iff \forall i,j(A_{i,j}\leq B_{i,j})$ under this order, what would the matrix $A$ be called in ...
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Why is the boolean "OR" operator denoted as "+"?

I learnt boolean algebra as part of a computer hardware course where the focus was very much on using it as a foundation for creating digital logic blocks out of gates, so there was very early on ...
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How can I define a module over a Boolean ring?

I want to give an example of a module over a Boolean ring $R$ (in particular, the power set of a set $X$ under symmetric difference and set intersection). I thought of defining a map $(Y, m) \...
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3 answers
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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. ...
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1 answer
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If $R$ is a Boolean ring which is in fact a field, then it must be isomorphic to $\mathbb Z / 2 \mathbb Z$ [duplicate]

Suppose $R$ is a Boolean ring ($\forall r \in R, r^2 =r$) which is in fact a field. Show it then must be isomorphic to $\mathbb Z / 2 \mathbb Z$. I already managed to show that every Boolean ring is ...
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Spectrum of a ring in which $a^p=a$ for all $a$ and prime number $p$

It is known the result that spectrum of a boolean ring is zerodimensional and compact topological space, e.g. has a base with clopen sets and satisfies Haussdorf condition. It is asked if this result ...
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Let R be a ring in which $x^2=x$ for all x ε R. (a) Prove that $x + x = 0$ for for all x ε R. (b) Prove that R is commutative. [duplicate]

I am aware this question has been asked many times but I'm still struggling to understand a few things. Let R be a ring in which $x^2=x$ for all x ε R. (a) Prove that $x + x = 0$ for for all x ε R. ...
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Irreducibility for boolean ring?

Every element in $x \in A$ in a commutative bolean ring $A$ is by definition equal to $xx$, and so therefore there are no irreducible elements in the usual sense. So I ask, how might one define ...
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How to test whether an element is in a principal ideal of a ring without enumerating all elements of the ring?

Suppose we have a finite, boolean ring $A$ induced by a finite, commutative, boolean monoid $X$ containg $0$ as in: this question. You only need the first few paragraphs of that long post. Suppose I ...
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What is an elegant way of defining complement of an element in this ring?

About the ring: $R$ is a commutative boolean ring with $1$ in which for each $x \in R$, $x + x = 0$. There is a partial order on elements of $R$ written $x \leqslant y \iff xy = y$. The ordering ...
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0 answers
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Finite Boolean ring with $1 \neq 0$ is isomorphic to $\mathbb{Z}_2\times \mathbb{Z}_2 \times \dots \times \mathbb{Z}_2$ [duplicate]

Let $R$ be a finite Boolean ring with $1 \neq 0$. Show that $R\cong \mathbb{Z}_2\times \mathbb{Z}_2 \times \dots \times \mathbb{Z}_2$. This is exercise 2 on p267 in the book abstract algebra of ...
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Commutativity of Boolean ring

I could not prove that "Every Boolean ring is commutative", but I found it on the Internet. I am giving the idea of the proof that I "learned". Let $R$ be the Boolean ring. We can easily prove that $...
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Equivalent properties of a proper ideal of a generalized boolean algebra

I do not understand the item c) of the following question, the exercise 9 from section 1.2 from the book "Lattice-ordered Rings and Modules" from Stuart A. Steinberg: A generalized boolean algebra is ...
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1 answer
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Proving associativity of a certain binary aperation in any complemented distributive lattice

If, in a Boolean lattice $(X,\vee,\wedge,0,1,')$ (i.e. a complemented distributive lattice), we define $x+y=(x'\wedge y)\vee(x\wedge y')$, is there an elegant way to prove that $(x+y)+z=x+(y+z)$ ...
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2 votes
1 answer
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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 ...
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1 answer
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Shall we call this relation an "isomorphism"?

This is a short question: Assume the values: $\mathbf{True}$, $\mathbf{False}$ and the logic symbols: $\land,\lor$ Is $\mathbf{True}$, $\mathbf{False}$ under $\land,\lor \,$ isomorphic to $\...
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7 votes
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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,...
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2 votes
1 answer
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GRE 9768 #60 Boolean non-commutative rings: Prove $(-s)^2=s^2$ without commutativity.

GRE 9768 #60 Ian Coley's approach is to prove $(I)$ and $(I) \implies (II) \implies (III)$ In proving $(I)$, how do we prove $$(-s)^2=s^2$$ without commutativity (but with $s=s^2$, if need ...
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GRE 9768 #60 1. Does $(s+t)^2=s^2+t^2$ imply $s+s=0$? 2. Idempotent matrices do not form a ring?

GRE 9768 #60 on what appears to be Boolean rings: Ian Coley's approach is to prove $(I)$ and $(I) \implies (II) \implies (III)$ I think $(II) \implies (I)$. My attempt: $$(s+t)^2=s^2+t^2 \...
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-1 votes
1 answer
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Set Ring and Algebraic Ring

Let $\mathscr{R}$ be a ring of sets and define set operations $\odot=\text{multiplication}$ and $\oplus=\text{addition}$ by $$E\odot F=E\cap F$$ and $$E\oplus F=E\triangle F$$ then $\mathscr{R}$ is ...
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1 vote
2 answers
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Boolean algebra/ring: Compute product of sum of combinations of variables

Let's say I have a set of variables $a_1, ... ,a_n \in \{0, 1\}$ and a boolean algebra where multiplication is $\land$ and sum is $\oplus$ (exclusive or). This is like in bit operations in computers. ...
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0 votes
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On Boolean algebras and Atoms

I know plenety of questions have been made around this topics, but i've got an specific one. If $\mathfrak A$ Is an infinite boolean algebra, and $C$ is a finite subset of atoms in $\mathfrak A$, then ...
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Are these boolean rings integral domains?

Let X be a set and let P(X) denote the Boolean ring whose elements are the subsets of X, with addition being symmetric difference and multiplication being intersection. Is P({1}) an integral domain? ...
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1 answer
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Show that a Boolean ring is a commutative ring. [closed]

I have a question about (https://math.stackexchange.com/q/10279)'s proof to this. I also asked as a comment but I am unsure whether it will be replied to since the post was made 8 years ago? My ...
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1 vote
1 answer
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How to prove convergence of a sequence of binary numbers

I have a boolean expression with 4 inputs and 1 output, that when iterated onto itself(output->input(s)), the function converges to 1. How do I go about proving the convergence of a sequence of binary ...
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2 votes
1 answer
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A non-Boolean ring without unity with this property

I'm exploring rings that have the following property: $\rule{10cm}{0.4pt}$ For all $x \in R$, and for all $n \in \mathbb{N}$, there exists $y \in R$ such that $\sum_{i=1}^{n} x = y \cdot x$. $\rule{...
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2 votes
1 answer
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Expressing real function algebraically for every point in its domain

Let's say that f(x) is a function with its domain called $A$ with its codomain labeled $B$. The indicator function $I_{\{p\}}(x)$ has a value of the multiplicative identity, $1$, when $x=p$, and $0$ ...
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1 answer
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Compliment multiplication Compliment

Since, $a+a'=1$ and $a.a'=0$ (Boolean Algebra complement laws), where $a$ is a variable and $a'$ is its complement. Can you please explain what $a'.a'=$ would be? Also, please prove your result as ...
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2 votes
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Boolean algebra : number of atoms, unions, intersections

Is there a formula that relates the number of atoms of a boolean algebra to the number of different unions/intersections of some generating elements? Consider the boolean sub-algebra $A$ generated by ...
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3 answers
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Question about Boolean rings.

As we know every finite boolean ring in which $1≠0$ is isomorphic to, $\mathbb{Z}_2 \times \mathbb{Z}_2 \times \cdots \times \mathbb{Z}_2$. So, is every infinite boolean ring is isomorphic to $\...
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