I'm reading the article 


*M. Hochster*, [**Prime ideal structure in commutative rings**](http://www.ams.org/mathscinet-getitem?mr=251026), *Trans. Amer. Math. Soc.* **142** (1969), 43--60. Freely available [here on the journal's website](http://www.ams.org/journals/tran/1969-142-00/S0002-9947-1969-0251026-X/).


But, I can not find the definition of the symbol "$[b\_1 , b\_2]$" in the proof of Theorem 4 (page 49, which is 7th page of the pdf). 


Could someone explain what this notation means there?
Possibly it is explained somewhere in the paper, but I could not locate it.



What does the notation $[b_1,b_2]$ in M. Hochster's "Prime Ideal Structure in Commutative Rings" mean?
My best guess for what $[b\_1,b\_2]$ is supposed to mean is the subring of $A$ generated by $b\_1$ and $b\_2$. However, even with this interpretation, there are some statements that aren't quite right (though as I recall, this particular paper has a lot of minor errors of this sort, so this shouldn't be too surprising). Here is how I would rewrite the final three sentences of the proof of Theorem 4:


Let $f:\mathbb{Z}[x\_1,x\_2]\to A$ be the unique homomorphism sending $x\_i$ to $b\_i$ and let $I=(x\_1,x\_2)\subset\mathbb{Z}[x\_1,x\_2]$. For any $z\in I$, $f(z)$ vanishes off of $d(b\_1)\cup d(b\_2)$, and its image restricted to $d(b\_1)\cup d(b\_2)$ is equal to its image restricted to the finite set $Y$. Thus we need only find $z\in I$ such that $z\not\in f^{-1}(\phi(y))$ for each $y\in Y$, and then $c=f(z)$ will be an element of $(b\_1,b\_2)$ such that $d(c)=d(b\_1)\cup d(b\_2)$. But each $f^{-1}(\phi(y))$ is a prime ideal in $\mathbb{Z}[x\_1,x\_2]$ which does not contain $I$ (since every point of $Y$ is in $d(b\_1)\cup d(b\_2)$), and there are only finitely many of them, so by prime avoidance such a $z$ must exist.


