Ring of Sets vs Ring in Universal Algebra

Ring of Sets by corresponding union to addition and intersection to multiplication ($\varnothing$ to $0$ and $E$ (universe of set) to $1$ as well) satisfy most ring axioms. However the fundamental difference is that it doesn't have an additive inverse because $X\cup Y=\varnothing$ if and only if $X=\varnothing$ and $Y=\varnothing$.

Thus your definition of Ring of Sets is not a ring rigorously.

Another way to define Ring of Sets is known as Boolean Ring, i.e by corresponding symmetric difference of sets to addition and intersection to multiplication ($\varnothing$ to $0$ and $E$ (universe of set) to $1$ as well). It satisfies all ring axioms with an additive inverse.

Please read this post for more detail.

Tags:

Abstract Algebra

Ring Theory

Universal Algebra

Elementary Set Theory

Boolean Ring

Related

Evaluate $\int\frac1{1+x^n}dx$ for $n\in\mathbb R$ Why does Tao use the word *metatheory* in this context? Why does Friedberg say that the role of the determinant is less central than in former times? Prove that $G$ is an abelian group if $\{(g, g):g\in G\}$ is a normal subgroup. There must exist a random variable with certain given law? If $p\geq 5$ is a prime number, show that $p^2+2$ is composite. Prove that for any set of 5 integers, there is at least one subset of 3 integers whose sum is divisible by 3 Any way to solve this right angle triangle problem without trig? Determinant of the derivative of a map between manifolds Gradient of Sobolev function on level sets Why does a unitary in the Calkin algebra always lift to an (co-)isometry? If $f$ is differentiable on $[1,2]$, then $\exists \alpha\in(1,2): f(2)-f(1) = \frac{\alpha^2}{2}f'(\alpha)$

Recent Posts