It was more than a year ago that I opened a package that I got in the mail, taking out this green ex-library hardcover in excellent condition. Now, I honestly can’t remember what prompted me to order it (perhaps it was the author’s name), but I remember reading the first few sections and feeling that it would be worthwhile to spend some time over its pages to learn more about those primes. Unfortunately, I didn’t manage to keep reading at the time, but a few months ago I decided to push through this volume with a little spare time I had, and this became my first real serious conversation with the zero divisor.

Certainly, no algebraist can ever escape the grasp of the zero divisor. In a ring $ R$, a nonzero element $ r\in R$ is called a zero divisor if $ rs = 0$ for some nonzero $ s\in R$. In even basic questions on ring theory, zero divisors are bound to be lurking. Our topic today is commutative rings, so we’ll assume from now on that $ R$ is commutative. The rings probably easiest to understand, at least if we’re not considering relations to other rings, are fields. If $ R$ is not a field, then it has an ideal $ I$ that is not prime so $ R/I$ already has zero divisors. So even if $ R$ is a domain, some of its quotients will not be as long as $ R$ is not a field.

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