Actually, i posted the exact same question before (about a year ago), but now i lost my past account so i couldn't find the past post..
So i googled this, but i couldn't find a satisfying post.
I'll illustrate two different situations below
Let $X$ be a set and $\sum \subset P(X)$. Then, $\sum$ is a sigma algebra on $X$ iff (1)it is closed under countable union, (2)it is closed under complement and (3)$X\in \sum$
If we start measure theory via this definition, just like topology, for a given sigma algebra, we can immediately know that on which set this sigma algebra is defined.
Thus, this definition makes it possible to view a sigma-algebra as a measurable space.
Let $X$ be a set and $\sum \subset P(X)$. Then $\sum$ is a sigma ring on $X$ Iff (i) it is closed under countable union, (ii)$\forall A,B\in \sum, A-B\in \sum$.
If we start measure theory in this way, when we want to talk about a measure space, a set and a sigma-ring on this set should be given together. This is the only disadvantage of sigma ring in my opinion.
Let $M=\bigcup \sum$.
Then, $\sum$ is a sigma algebra on $M$ iff $M\in \sum$.
Thus, the definition of sigma ring is strictly stronger than that of sigma algebra.
I cannot understand why mathematicians got rid of this generalized definition and prefer a weaker one.
I remember that someone answered me that a lot of interesting spaces are integrable themselves. Which means, lot of interesting $\sigma$-whatsoever contains the whole space, so they are $\sigma$-algebras.
But, isn't there any interesting $\sigma$-ring which is not $\sigma$-algebra?
I really hate to go back and define something again more generally so that i have to prove every single theorem depending on property of the older definition.
For example, i started analysis with Rudin-PMA and he defined Topology as topology induced by metric space in usual sense. It was painful to me to distinguish theorems which hold only in metric space and which hold in topological space, when i learned general-topology later.
It's hard to confirm myself why mathematicians prefer this weak definition taking this risk..