# 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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1 vote
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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 ...
1 vote
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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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1 vote
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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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1 vote
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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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### 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 $(x+y)+z=x+(y+z)$ ...
88 views

### 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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### 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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### 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 ...
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 ...
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 \$\...