What does “factor out” an ideal mean?

Question

I am studying this article. In proving Theorem 2.14—

Let $R$ be an $\alpha$-ring and $S=R[x,\alpha]$. Then $S$ is $\alpha$-Jacobson if and only if $R$ is $\alpha$-Jacobson

—the author says that:

. . . we can factor out $(P \cap R)S $, which is contained in $P$, and assume that $P \cap R = 0$

(where $P$ is an $\alpha$-prime ideal of $S$).

I don't understand what this “factor out” means. Any help would be great.

Answer 1

It means to consider the quotient. When you do $P/(P\cap R)$, now $P\cap R$ is the zero of the quotient.