# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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 ...
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### A Boolean ring in which if $2a=0$ then $a=0$

In every Boolean ring we have $a^2=a$ for every $a$ in the ring. In some Boolean rings, if $2a=0$ then $a=0$. How to show this ring has just one member? Thanks in advance
Let $2^{\Bbb N}$ be the set of infinite binary sequences $\{x_n\}$ where $x_n\in \{0,1\}$ for every $n\in \Bbb N$. I want it to fit the axioms of a known algebraic structure such that the following ...
Can a Boolean ring be somehow extended to infinite binary sequences? If $2^{\Bbb N}$ is the Cantor space (the set of all infinite binary sequences), is it a ring if the operations are defined term by ...